# The Monoid Consisting of Kuratowski Operations

Authors: Szymon Plewik; Marta Walczyńska
Journal: Journal of Mathematics
 (2013)

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## Abstract

The paper fills gaps in knowledge about Kuratowski
operations which are already in the literature. The Cayley table
for these operations has been drawn up. Techniques, using only
paper and pencil, to point out all semigroups and its isomorphism
types are applied. Some results apply only to topology, and one cannot bring them out, using only properties of the complement and
a closure-like operation. The arguments are by systematic study
of possibilities.


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## Body

## 1. Introduction

LetX be a topological space. Denote by A- closure of the set A⊆X. Let Ac be the complement of A; that is, X∖A=Ac. The aim of this paper is to examine monoids generated under compositions from the closure and the complement. A widely known fact due to Kuratowski [1] states that at most 14 distinct operations can be formed from such compositions. Mark them as follows. Kuratowski operations:σ0(A)=A (the identity),σ1(A)=Ac (the complement),σ2(A)=A-  (the closure),σ3(A)=Ac-,σ4(A)=A-c,σ5(A)=Ac-c(the interior),σ6(A)=A-c-,σ7(A)=Ac-c-,σ8(A)=A-c-c,σ9(A)=Ac-c-c,σ10(A)=A-c-c-,σ11(A)=Ac-c-c-,σ12(A)=A-c-c-c,σ13(A)=Ac-c-c-c.The following rules was found in the original paper by Kuratowski [1, pages 183-184]. In the Engelking book [2], they are commented by a hint on page 81. Cancellation rules:
(1)A-c-=A-c-c-c-,Ac-c-=Ac-c-c-c-.Kuratowski operations have been studied by several authors, for example, [3] or [4]. A list of some other authors one can find in the paper [5] by Gardner and Jackson. For the first time these operations were systematically studied in the dissertation by Kuratowski, whose results were published in [1]. Tasks relating to these operations are usually resolved at lectures or exercises with general topology. They are normally left to students for independent resolution. For example, determine how many different ways they convert a given set.This note is organized as follows. Kuratowski operations and their marking are described in the introduction. Their properties of a much broader context than for topologies are presented in Section2. The Cayley table, for the monoid 𝕄 of all Kuratowski operations, has been drawn up in Section 3. We hope that this table has not yet been published in the literature. Having this table, one can create a computer program that calculates all the semigroups contained in 𝕄. However, in Sections 4–8, we present a framework (i.e., techniques using only paper and pencil) to point out all 118 semigroups and 56 isomorphism types of them. The list of 43 semigroups which are not monoids is presented in Section 9. In this section, also isomorphism types are discussed in order of the number of elements in semigroups. Finally, we present cancellation rules (relations) motivated by some topological spaces.


## 2. Cancellation Rules

A mapf:P(X)→P(X) is called: (i)
increasing, if A⊆B implies f(A)⊆f(B);(ii)
decreasing, if A⊆B implies f(B)⊆f(A);(iii)
an involution, if the composition f∘f is the identity;(iv)
an idempotent, if f∘f=f. Assume that A↦σ0(A) is the identity, A↦σ1(A) is a decreasing involution, and A↦σ2(A) is an increasing idempotent map. Other operations σi let be compositions of σ1 and σ2 as it was with the kuratowski operations. We get the following cancellation rules.Lemma 1.
IfB⊆σ2(B), then
(2)σ2∘σ12=σ6,σ2∘σ13=σ7.Proof.
For clarity of this proof, use designationsσ1(A)=Ac and σ2(A)=A-. Thus, we shall prove
(3)A-c-c-c-=A-c-,Ac-c-c-c-=Ac-c-.
We start with A-c-c⊆A-c-c-, substituting B=A-c-c in B⊆B-. This corresponds to A-c-c-c⊆A-c-cc=A-c-, since σ1 is a decreasing involution. Hence A-c-c-c-⊆A-c-, since σ2 an increasing idempotent.
Sinceσ1 is decreasing, σ2 is increasing, and B⊆B-, we have
(4)B-c⊆Bc⊆Bc-.
Thus A-c-c⊆A-cc-=A-, if we put A-c=B. Again using that σ2 is increasing and σ1 is decreasing, we obtain A-c-c-⊆A- and then A-c⊆A-c-c-c. Finally, we get A-c-⊆A-c-c-c-.
WithAc in the place of A in the rule A-c-=A-c-c-c- we get the second rule.In academic textbooks of general topology, for example, [2, Problem 1.7.1.], one can find a hint suggested to prove the above cancellation rules. Students go like this: steps A⊆A- and Ac⊆Ac- lead to Ac-c⊆A; the special case A-c-c-c⊆A-c- (of Ac-c⊆A) leads to A-c-c-c-⊆A-c-; steps A-c-c⊆A- and A-c-c-⊆A- lead to A-c-⊆A-c-c-c-; at the end, use the last step of the proof of Lemma 1. Note that, the above proofs do not use the axioms of topology as follows: (i)
∅=∅-;(ii)
(A∪B)-=A-∪B-. In the literature there are articles in which Kuratowski operations are replaced by some other mappings. For example, Koenen [6] considered linear spaces and put σ2(A) to be the convex hull of A. In fact, Shum [7] considered σ2 as the closure due to the algebraic operations. Add to this, that these operations can be applied to so-called Fréchet (V) spaces, which were considered in the book [8, pages 3–37].


## 3. The Monoid𝕄

Let𝕄 be the monoid consisting of all Kuratowski operations; that is, there are assumed cancellation rules: σ6=σ2∘σ12 and σ7=σ2∘σ13. Fill in the Cayley table for 𝕄, where the row and column marked by the identity are omitted. Similarly as in [9], the factor that labels the row comes first, and that the factor that labels the column is second. For example, σi∘σk is in the row marked by σi and the column marked by σk(5)σ1σ2σ3σ4σ5σ6σ7σ8σ9σ10σ11σ12σ13σ1σ0σ4σ5σ2σ3σ8σ9σ6σ7σ12σ13σ10σ11σ2σ3σ2σ3σ6σ7σ6σ7σ10σ11σ10σ11σ6σ7σ3σ2σ6σ7σ2σ3σ10σ11σ6σ7σ6σ7σ10σ11σ4σ5σ4σ5σ8σ9σ8σ9σ12σ13σ12σ13σ8σ9σ5σ4σ8σ9σ4σ5σ12σ13σ8σ9σ8σ9σ12σ13σ6σ7σ6σ7σ10σ11σ10σ11σ6σ7σ6σ7σ10σ11σ7σ6σ10σ11σ6σ7σ6σ7σ10σ11σ10σ11σ6σ7σ8σ9σ8σ9σ12σ13σ12σ13σ8σ9σ8σ9σ12σ13σ9σ8σ12σ13σ8σ9σ8σ9σ12σ13σ12σ13σ8σ9σ10σ11σ10σ11σ6σ7σ6σ7σ10σ11σ10σ11σ6σ7σ11σ10σ6σ7σ10σ11σ10σ11σ6σ7σ6σ7σ10σ11σ12σ13σ12σ13σ8σ9σ8σ9σ12σ13σ12σ13σ8σ9σ13σ12σ8σ9σ12σ13σ12σ13σ8σ9σ8σ9σ12σ13It turns out that the above table allows us to describe all semigroups contained in𝕄, using pencil-and-paper techniques, only. The argument will be by a systematic study of possibilities. Preparing the list of all semigroups consisting of Kuratowski operations we used following principles: (i)
minimal collection of generators is written using〈A,B,…,Z〉, where letters denote generators;(ii)
when a semigroup has a few minimal collections of generators, then its name is the first collection in the dictionary order;(iii)
all minimal collections of generators are written with the exception of some containingσ0;(iv)
we leave to the readers verification that our list is complete, sometimes we add hints.


## 4. Semigroups withσ1

Observe that each semigroup which containsσ0 is a monoid. Since σ1∘σ1=σ0, a semigroup which contains σ1 is a monoid, too.Theorem 2.
There are three monoids containingσ1: (1)
〈σ1〉={σ0,σ1};(2)
𝕄=〈σ1,σi〉={σ0,σ1,…,σ13}, where i∈{2,3,4,5};(3)
let𝕄1=〈σ1,σ6〉. If j∈{6,7,…,13}, then
(6)𝕄1=〈σ1,σj〉={σ0,σ1}∪{σ6,σ7,…,σ13}.Proof.
The equality〈σ1〉={σ0,σ1} is obvious.
Sinceσ2=σ3∘σ1=σ1∘σ4=σ1∘σ5∘σ1, we have 〈σ1,σ2〉=〈σ1,σ3〉=〈σ1,σ4〉=〈σ1,σ5〉=𝕄.
Ifj∈{6,7,…,13}, then any composition σj∘σi or σk∘σj belongs to 𝕄1, and so 〈σ1,σj〉⊆𝕄1. We have σ7=σ6∘σ1, σ8=σ1∘σ6, σ9=σ1∘σ7, σ10=σ6∘σ6, σ11=σ6∘σ7, σ12=σ8∘σ6, and σ13=σ8∘σ7, and so 𝕄1=〈σ1,σ6〉={σ0,σ1}∪{σ6,σ7,…,σ13}.
Sinceσ6=σ7∘σ1=σ1∘σ8=σ1∘σ9∘σ1=σ10∘σ1∘σ10=σ11∘σ11∘σ1=σ1∘σ12∘σ12 and σ6=σ1∘σ13∘σ1∘σ13∘σ1, we have 〈σ1,σj〉=𝕄1 for j∈{6,7,…,13}.Consider the permutation(7)(σ0σ1σ2σ3σ4σ5σ6σ7σ8σ9σ10σ11σ12σ13σ0σ1σ5σ4σ3σ2σ9σ8σ7σ6σ13σ12σ11σ10).
It determines an automorphism 𝔸:𝕄→𝕄.Theorem 3.
The identity and𝔸 are the only automorphisms of 𝕄.Proof.
Delete rows and columns marked byσ1 in the Cayley table for 𝕄. Then, check that the operation σ3 is in the row or the column marked by σ3 only. Also, the operation σ4 is in the row or the column marked by σ4 only. Therefore the semigroup
(8)〈σ3,σ4〉={σ2,σ3,…,σ13}
has a unique minimal set of generators {σ3,σ4}. The reader is left to check this with the Cayley table for 𝕄.
Suppose𝔾 is an automorphism of 𝕄. By Theorem 2, 𝔾 transforms the set {σ2,σ3,σ4,σ5} onto itself. However σ2 and σ5 are idempotents, but σ3 and σ4 are not idempotents. So, there are two possibilities: 𝔾(σ3)=σ3 and 𝔾(σ4)=σ4, which implies that 𝔾 is the identity; 𝔾(σ3)=σ4 and 𝔾(σ4)=σ3, which implies 𝔾=𝔸. The reader is left to check this with the Cayley table for 𝕄. We offer hints: σ2=σ3∘σ4, σ5=σ4∘σ3, σ6=σ3∘σ2, σ7=σ3∘σ3, σ8=σ4∘σ4, σ9=σ4∘σ5, σ10=σ6∘σ6, σ11=σ6∘σ7, σ12=σ8∘σ6, and σ13=σ8∘σ7, to verify the details of this proof.


## 5. The Monoid of All Idempotents

The set{σ0,σ2,σ5,σ7,σ8,σ10,σ13} consists of all squares in 𝕄. These squares are idempotents and lie on the main diagonal in the Cayley table for 𝕄. They constitute a monoid and
(9){σ0,σ2,σ5,σ7,σ8,σ10,σ13}=〈σ0,σ2,σ5〉.
The permutation
(10)(σ0σ2σ5σ7σ8σ10σ13σ0σ2σ5σ8σ7σ10σ13)
determines the bijection 𝕀:〈σ0,σ2,σ5〉→〈σ0,σ2,σ5〉, such that
(11)𝕀(α∘β)=𝕀(β)∘𝕀(α),
for any α,β∈〈σ0,σ2,σ5〉. To verify this, apply equalities σ2∘σ5=σ7, σ5∘σ2=σ8, σ2∘σ5∘σ2=σ10, and σ5∘σ2∘σ5=σ13. Any bijection 𝕀:G→H, having property 𝕀(α∘β)=𝕀(β)∘𝕀(α), transposes Cayley tables for semigroups G and H. Several semigroups contained in 𝕄 have this property. We leave the reader to verify this.We shall classify all semigroups contained in the semigroup〈σ2,σ5〉. Every such semigroup can be extended to a monoid by attaching σ0 to it. This gives a complete classification of all semigroups in 〈σ0,σ2,σ5〉.The semigroup〈σ2,σ5〉 contains six groups with exactly one element.Semigroups〈σ2,σ10〉={σ2,σ10} and 〈σ5,σ13〉={σ5,σ13} are monoids. Both consist of exactly two elements and are not groups, so they are isomorphic.Semigroups〈σ7,σ10〉 and 〈σ8,σ13〉 are isomorphic, in particular 𝔸[〈σ7,σ10〉]=〈σ8,σ13〉. Also, semigroups 〈σ7,σ13〉 and 〈σ8,σ10〉 are isomorphic by 𝔸. Every of these four semigroups has exactly two elements. None of them is a monoid. They form two types of nonisomorphic semigroups, because the bijections {(σ7,σ10),(σ10,σ8)} and {(σ7,σ8),(σ10,σ10)} are not isomorphisms.Semigroups〈σ2,σ7〉={σ2,σ7,σ10} and 〈σ5,σ8〉={σ5,σ8,σ13} are isomorphic. Also, semigroups 〈σ2,σ8〉={σ2,σ8,σ10} and 〈σ5,σ7〉={σ5,σ7,σ13} are isomorphic. In fact, 𝔸[〈σ2,σ7〉]=〈σ5,σ8〉 and 𝔸[〈σ2,σ8〉]=〈σ5,σ7〉. None of these semigroups is a monoid. They form two types of nonisomorphic semigroups. Indeed, any isomorphism between 〈σ2,σ7〉 and 〈σ2,σ8〉 must be the identity on the monoid 〈σ2,σ10〉. Therefore would have to be the restriction of 𝕀. But 𝕀 restricted to 〈σ7,σ10〉 is not an isomorphism.Semigroups〈σ2,σ13〉=〈σ2,σ7,σ8〉={σ2,σ7,σ8,σ10,σ13} and 〈σ5,σ10〉=〈σ5,σ7,σ8〉={σ5,σ7,σ8,σ10,σ13} are not monoids. They are isomorphic by 𝔸.The semigroup〈σ2,σ5〉 contains exactly one semigroup with four elements 〈σ7,σ8〉=〈σ10,σ13〉={σ7,σ8,σ10,σ13} which is not a monoid.Note that〈σ2,σ5〉 contains twenty different semigroups with nine non-isomorphism types. These are six isomorphic groups with exactly one element 〈σ2〉≅〈σ5〉≅〈σ7〉≅〈σ8〉≅〈σ10〉≅〈σ13〉, two isomorphic monoids with exactly two elements 〈σ2,σ10〉≅〈σ5,σ13〉, two pairs of isomorphic semigroups with exactly two elements 〈σ7,σ10〉≅〈σ8,σ13〉 and 〈σ7,σ13〉≅〈σ8,σ10〉, two pairs of isomorphic semigroups with exactly three elements 〈σ2,σ7〉≅〈σ5,σ8〉 and 〈σ2,σ8〉≅〈σ5,σ7〉, a semigroup with exactly four elements 〈σ7,σ8〉, and two isomorphic semigroups with exactly five elements 〈σ2,σ13〉≅〈σ5,σ10〉 and also 〈σ2,σ5〉. Thus, 〈σ2,σ5〉 contains twenty different semigroups with nine isomorphism types. But 〈σ0,σ2,σ5〉 contains forty-one different semigroups with seventeen isomorphism types. Indeed, adding σ0 to semigroups contained in 〈σ2,σ5〉, which are not monoids, we get twenty monoids with eight non-isomorphism types. Adding σ0 to a group contained in 〈σ2,σ5〉 we get a monoid isomorphic to 〈σ2,σ10〉.


## 6. The Semigroup Consisting of{σ6,σ7,…,σ13}

Using the Cayley table for𝕄, check that
(12)𝔸[{σ6,σ7,…,σ13}]={σ6,σ7,…,σ13}.
Similarly, check that the semigroup 〈σ6,σ9〉={σ6,σ7,…,σ13} can be represented as 〈σ6,σ13〉, 〈σ7,σ12〉, 〈σ8,σ11〉, 〈σ9,σ10〉, or 〈σ11,σ12〉. Also 〈σ6,σ9〉=〈σ6,σ7,σ8〉=〈σ7,σ8,σ9〉=〈σ10,σ11,σ13〉=〈σ10,σ12,σ13〉. These representations exhaust all minimal collections of the Kuratowski operations which generate 〈σ6,σ9〉. Other semigroups included in 〈σ6,σ9〉 have one or two minimal collection of generators. One generator has groups 〈σ6〉={σ6,σ10}, 〈σ9〉={σ9,σ13}, 〈σ11〉={σ7,σ11}, and 〈σ12〉={σ8,σ12}. Each of them has exactly two elements, so they are isomorphic. Semigroups 〈σ7,σ10〉, 〈σ7,σ13〉, 〈σ8,σ10〉, 〈σ8,σ13〉, and 〈σ7,σ8〉 are discussed in the previous section. Contained in 〈σ6,σ9〉 and not previously discussed semigroups are 〈σ6,σ7〉, 〈σ6,σ8〉,〈σ7,σ9〉, and 〈σ8,σ9〉. We leave the reader to verify that the following are all possible pairs of Kuratowski operations which constitute a minimal collection of generators for semigroups contained in 〈σ6,σ9〉, but different from the whole. One has (i)
〈σ6,σ7〉=〈σ6,σ11〉=〈σ10,σ11〉={σ6,σ7,σ10,σ11};(ii)
〈σ6,σ8〉=〈σ6,σ12〉=〈σ10,σ12〉={σ6,σ8,σ10,σ12};(iii)
〈σ7,σ9〉=〈σ9,σ11〉=〈σ11,σ13〉={σ7,σ9,σ11,σ13};(iv)
〈σ8,σ9〉=〈σ9,σ12〉=〈σ12,σ13〉={σ8,σ9,σ12,σ13}.Proposition 4.
Semigroups〈σ6,σ7〉 and 〈σ8,σ9〉 are isomorphic, and also semigroups 〈σ6,σ8〉 and 〈σ7,σ9〉 are isomorphic, but semigroups 〈σ6,σ7〉, 〈σ6,σ8〉 are not isomorphic.Proof.
Isomorphisms are defined by𝔸. Suppose J:〈σ6,σ7〉→〈σ6,σ8〉 is an isomorphism. Thus J[〈σ7,σ10〉]=〈σ8,σ10〉. Given J(σ7)=σ8, we get σ10=J(σ10)=J(σ7∘σ10)≠σ8∘σ10=σ8. But J(σ7)=σ10 implies σ10=J(σ7)=J(σ10∘σ7)≠σ8∘σ10=σ8. Both possibilities lead to a contradiction.So,〈σ6,σ9〉 contains eighteen different semigroups with eight non-isomorphism types. Indeed, these are four isomorphic groups with exactly one element 〈σ7〉≅〈σ8〉≅〈σ10〉≅〈σ13〉, four isomorphic groups with exactly two elements 〈σ6〉≅〈σ9〉≅〈σ11〉≅〈σ12〉, two pairs of isomorphic semigroups with exactly two elements 〈σ7,σ10〉≅〈σ8,σ13〉 and 〈σ7,σ13〉≅〈σ8,σ10〉, and five semigroups with exactly four elements 〈σ6,σ7〉≅〈σ8,σ9〉, 〈σ6,σ8〉≅〈σ7,σ9〉,  〈σ7,σ8〉, and also 〈σ6,σ9〉.


## 7. Remaining Semigroups in〈σ3,σ4〉

We have yet to discuss semigroups included in〈σ3,σ4〉, not included in 〈σ2,σ5〉 and containing at least one of Kuratowski operation σ2, σ3, σ4, or σ5. It will be discussed up to the isomorphism 𝔸. Obviously, 〈σ2〉={σ2} and 〈σ5〉={σ5} are groups.


### 7.1. Extensions of〈σ2〉 and 〈σ5〉 with Elements of 〈σ6,σ9〉

Monoids〈σ2,σ6〉={σ2,σ6,σ10} and 〈σ2,σ10〉={σ2,σ10} have different numbers of elements. Also, 〈σ2,σ6〉 is isomorphic to 〈σ0,σ6〉. Nonisomorphic semigroups 〈σ2,σ7〉 and 〈σ2,σ8〉 are discussed above. The following three semigroups: (i)
〈σ2,σ11〉=〈σ2,σ6,σ7〉={σ2,σ6,σ7,σ10,σ11},(ii)
〈σ2,σ12〉=〈σ2,σ6,σ8〉={σ2,σ6,σ8,σ10,σ12},(iii)
〈σ2,σ13〉=〈σ2,σ7,σ8〉={σ2,σ7,σ8,σ10,σ13} are not monoids. They are not isomorphic. Indeed, any isomorphism between these semigroups would lead an isomorphism between 〈σ6,σ7〉, 〈σ6,σ8〉, or 〈σ7,σ8〉. This is impossible, by Proposition 4 and because 〈σ7,σ8〉 consists of idempotents, but σ6 is not an idempotent. The nine-element semigroup on the set {σ2,σ6,σ7,…,σ13} is represented as 〈σ2,σ9〉. So, we have added four new semigroups, which are not isomorphic with the semigroups previously discussed. These are 〈σ2,σ11〉, 〈σ2,σ12〉, 〈σ2,σ13〉, and 〈σ2,σ9〉.Using𝔸, we have described eight semigroups—each one isomorphic to a semigroup previously discussed—which contains σ5 and elements (at least one) of 〈σ6,σ9〉. Collections of generators: 〈σ2,σ6,σ13〉, 〈σ2,σ7,σ12〉, 〈σ2,σ8,σ11〉, 〈σ2,σ11,σ12〉, 〈σ2,σ11,σ13〉, and 〈σ2,σ12,σ13〉 are minimal in 〈σ2,σ9〉. Also, collections of generators: 〈σ5,σ7,σ12〉, 〈σ5,σ8,σ11〉, 〈σ5,σ9,σ10〉, 〈σ5,σ10,σ11〉, 〈σ5,σ10,σ12〉, and 〈σ5,σ11,σ12〉 are minimal in 〈σ5,σ6〉.


### 7.2. Extensions of〈σ3〉 and 〈σ4〉 by Elements from the Semigroup 〈σ6,σ9〉

Semigroups〈σ3〉={σ3,σ7,σ11} and 〈σ4〉={σ4,σ8,σ12} are isomorphic by 𝔸. They are not monoids. The semigroup 〈σ3〉 can be extended using elements of 〈σ6,σ9〉, in three following ways:(i)
〈σ3,σ6〉=〈σ3,σ10〉={σ3,σ6,σ7,σ10,σ11};(ii)
〈σ3,σ8〉=〈σ3,σ12〉={σ3,σ6,σ7,…,σ13};(iii)
〈σ3,σ9〉=〈σ3,σ13〉={σ3,σ7,σ9,σ11,σ13}.Semigroups 〈σ3,σ6〉 and 〈σ3,σ9〉 are not isomorphic. Indeed, suppose J:〈σ3,σ6〉→〈σ3,σ9〉 is an isomorphism. Thus, J is the identity on 〈σ3〉, J(σ6)=σ9, and J(σ10)=σ13. This gives a contradiction, since σ3∘σ6=σ10 and σ3∘σ9=σ7.Neither〈σ3,σ6〉 nor 〈σ3,σ9〉 has a minimal collection of generators with three elements, so they give new isomorphism types. Also, 〈σ2,σ9〉 is not isomorphic to 〈σ3,σ8〉, since 〈σ2,σ9〉 has a unique pair of generators and 〈σ3,σ8〉=〈σ3,σ12〉.Using𝔸, we get—isomorphic to previously discussed ones—semigroups 〈σ4,σ9〉=〈σ4,σ13〉={σ4,σ8,σ9σ12,σ13}, 〈σ4,σ6〉=〈σ4,σ10〉={σ4,σ6,σ8σ10,σ12}, and 〈σ4,σ7〉=〈σ4,σ11〉={σ4,σ6,σ7,…,σ13}. There exist minimal collections of generators, such as follows:
(13)〈σ3,σ8〉=〈σ3,σ6,σ9〉=〈σ3,σ6,σ13〉=〈σ3,σ9,σ10〉=〈σ3,σ10,σ13〉,〈σ4,σ7〉=〈σ4,σ6,σ9〉=〈σ4,σ9,σ10〉=〈σ4,σ6,σ13〉=〈σ4,σ10,σ13〉.


### 7.3. More Generators from the Set{σ2,σ3,σ4,σ5}

Now we check that〈σ2,σ3〉={σ2,σ3,σ6,σ7,σ10σ11} and 〈σ4,σ5〉={σ4,σ5,σ8,σ9,σ12σ13}=𝔸[〈σ2,σ3〉] and also 〈σ2,σ4〉={σ2,σ4,σ6,σ8,σ10σ12} and 〈σ3,σ5〉={σ3,σ5,σ7,σ9,σ11σ13}=𝔸[〈σ2,σ4〉] are two pairs of isomorphic semigroups which give two new isomorphism types. Each of these semigroups has six element, so in 𝕄 there are five six-element semigroups of three isomorphism types, since 〈σ2,σ5〉 has 6 elements which are idempotents.In〈σ3,σ4〉 there are seven semigroups which have three generators and have not two generators. These are (1)
〈σ2,σ3,σ8〉=〈σ2,σ3,σ9〉=〈σ2,σ3,σ12〉=〈σ2,σ3,σ13〉;(2)
〈σ4,σ5,σ6〉=〈σ4,σ5,σ7〉=〈σ4,σ5,σ10〉=〈σ4,σ5,σ11〉;(3)
〈σ2,σ4,σ7〉=〈σ2,σ4,σ9〉=〈σ2,σ4,σ11〉=〈σ2,σ4,σ13〉;(4)
〈σ3,σ5,σ6〉=〈σ3,σ5,σ8〉=〈σ3,σ5,σ10〉=〈σ3,σ5,σ12〉;(5)
〈σ2,σ5,σ6〉=〈σ2,σ5,σ9〉=〈σ2,σ5,σ11〉=〈σ2,σ5,σ12〉;(6)
〈σ2,σ3,σ5〉;(7)
〈σ2,σ4,σ5〉.


## 7.1. Extensions of〈σ2〉 and 〈σ5〉 with Elements of 〈σ6,σ9〉

Monoids〈σ2,σ6〉={σ2,σ6,σ10} and 〈σ2,σ10〉={σ2,σ10} have different numbers of elements. Also, 〈σ2,σ6〉 is isomorphic to 〈σ0,σ6〉. Nonisomorphic semigroups 〈σ2,σ7〉 and 〈σ2,σ8〉 are discussed above. The following three semigroups: (i)
〈σ2,σ11〉=〈σ2,σ6,σ7〉={σ2,σ6,σ7,σ10,σ11},(ii)
〈σ2,σ12〉=〈σ2,σ6,σ8〉={σ2,σ6,σ8,σ10,σ12},(iii)
〈σ2,σ13〉=〈σ2,σ7,σ8〉={σ2,σ7,σ8,σ10,σ13} are not monoids. They are not isomorphic. Indeed, any isomorphism between these semigroups would lead an isomorphism between 〈σ6,σ7〉, 〈σ6,σ8〉, or 〈σ7,σ8〉. This is impossible, by Proposition 4 and because 〈σ7,σ8〉 consists of idempotents, but σ6 is not an idempotent. The nine-element semigroup on the set {σ2,σ6,σ7,…,σ13} is represented as 〈σ2,σ9〉. So, we have added four new semigroups, which are not isomorphic with the semigroups previously discussed. These are 〈σ2,σ11〉, 〈σ2,σ12〉, 〈σ2,σ13〉, and 〈σ2,σ9〉.Using𝔸, we have described eight semigroups—each one isomorphic to a semigroup previously discussed—which contains σ5 and elements (at least one) of 〈σ6,σ9〉. Collections of generators: 〈σ2,σ6,σ13〉, 〈σ2,σ7,σ12〉, 〈σ2,σ8,σ11〉, 〈σ2,σ11,σ12〉, 〈σ2,σ11,σ13〉, and 〈σ2,σ12,σ13〉 are minimal in 〈σ2,σ9〉. Also, collections of generators: 〈σ5,σ7,σ12〉, 〈σ5,σ8,σ11〉, 〈σ5,σ9,σ10〉, 〈σ5,σ10,σ11〉, 〈σ5,σ10,σ12〉, and 〈σ5,σ11,σ12〉 are minimal in 〈σ5,σ6〉.


## 7.2. Extensions of〈σ3〉 and 〈σ4〉 by Elements from the Semigroup 〈σ6,σ9〉

Semigroups〈σ3〉={σ3,σ7,σ11} and 〈σ4〉={σ4,σ8,σ12} are isomorphic by 𝔸. They are not monoids. The semigroup 〈σ3〉 can be extended using elements of 〈σ6,σ9〉, in three following ways:(i)
〈σ3,σ6〉=〈σ3,σ10〉={σ3,σ6,σ7,σ10,σ11};(ii)
〈σ3,σ8〉=〈σ3,σ12〉={σ3,σ6,σ7,…,σ13};(iii)
〈σ3,σ9〉=〈σ3,σ13〉={σ3,σ7,σ9,σ11,σ13}.Semigroups 〈σ3,σ6〉 and 〈σ3,σ9〉 are not isomorphic. Indeed, suppose J:〈σ3,σ6〉→〈σ3,σ9〉 is an isomorphism. Thus, J is the identity on 〈σ3〉, J(σ6)=σ9, and J(σ10)=σ13. This gives a contradiction, since σ3∘σ6=σ10 and σ3∘σ9=σ7.Neither〈σ3,σ6〉 nor 〈σ3,σ9〉 has a minimal collection of generators with three elements, so they give new isomorphism types. Also, 〈σ2,σ9〉 is not isomorphic to 〈σ3,σ8〉, since 〈σ2,σ9〉 has a unique pair of generators and 〈σ3,σ8〉=〈σ3,σ12〉.Using𝔸, we get—isomorphic to previously discussed ones—semigroups 〈σ4,σ9〉=〈σ4,σ13〉={σ4,σ8,σ9σ12,σ13}, 〈σ4,σ6〉=〈σ4,σ10〉={σ4,σ6,σ8σ10,σ12}, and 〈σ4,σ7〉=〈σ4,σ11〉={σ4,σ6,σ7,…,σ13}. There exist minimal collections of generators, such as follows:
(13)〈σ3,σ8〉=〈σ3,σ6,σ9〉=〈σ3,σ6,σ13〉=〈σ3,σ9,σ10〉=〈σ3,σ10,σ13〉,〈σ4,σ7〉=〈σ4,σ6,σ9〉=〈σ4,σ9,σ10〉=〈σ4,σ6,σ13〉=〈σ4,σ10,σ13〉.


## 7.3. More Generators from the Set{σ2,σ3,σ4,σ5}

Now we check that〈σ2,σ3〉={σ2,σ3,σ6,σ7,σ10σ11} and 〈σ4,σ5〉={σ4,σ5,σ8,σ9,σ12σ13}=𝔸[〈σ2,σ3〉] and also 〈σ2,σ4〉={σ2,σ4,σ6,σ8,σ10σ12} and 〈σ3,σ5〉={σ3,σ5,σ7,σ9,σ11σ13}=𝔸[〈σ2,σ4〉] are two pairs of isomorphic semigroups which give two new isomorphism types. Each of these semigroups has six element, so in 𝕄 there are five six-element semigroups of three isomorphism types, since 〈σ2,σ5〉 has 6 elements which are idempotents.In〈σ3,σ4〉 there are seven semigroups which have three generators and have not two generators. These are (1)
〈σ2,σ3,σ8〉=〈σ2,σ3,σ9〉=〈σ2,σ3,σ12〉=〈σ2,σ3,σ13〉;(2)
〈σ4,σ5,σ6〉=〈σ4,σ5,σ7〉=〈σ4,σ5,σ10〉=〈σ4,σ5,σ11〉;(3)
〈σ2,σ4,σ7〉=〈σ2,σ4,σ9〉=〈σ2,σ4,σ11〉=〈σ2,σ4,σ13〉;(4)
〈σ3,σ5,σ6〉=〈σ3,σ5,σ8〉=〈σ3,σ5,σ10〉=〈σ3,σ5,σ12〉;(5)
〈σ2,σ5,σ6〉=〈σ2,σ5,σ9〉=〈σ2,σ5,σ11〉=〈σ2,σ5,σ12〉;(6)
〈σ2,σ3,σ5〉;(7)
〈σ2,σ4,σ5〉.


## 8. Semigroups which Are Contained in〈σ3,σ4〉

Groups〈σ2〉≅〈σ5〉≅〈σ7〉≅〈σ8〉≅〈σ10〉≅〈σ13〉 have one element and are isomorphic.Groups〈σ6〉≅〈σ9〉≅〈σ11〉≅〈σ12〉, monoids 〈σ2,σ10〉≅〈σ5,σ13〉, and also semigroups 〈σ7,σ10〉≅〈σ8,σ13〉 and 〈σ7,σ13〉≅〈σ8,σ10〉 have two elements and Cayley tables as follows:
(14)ABABABABABAABBBAABAABBABABAAABBBMonoids〈σ2,σ6〉≅〈σ5,σ9〉, and also semigroups 〈σ2,σ7〉≅〈σ5,σ8〉, 〈σ2,σ8〉≅〈σ5,σ7〉, and 〈σ3〉≅〈σ4〉 have three elements and Cayley tables as follows:
(15)ABCAABCBBCBCCBCABCAABCBCBCCCBCABCAACCBBBBCCCCABCABCBBCBCCBCBSemigroups〈σ6,σ7〉≅〈σ8,σ9〉, 〈σ6,σ8〉≅〈σ7,σ9〉, and 〈σ7,σ8〉 have four elements and Cayley tables as follows:
(16)ABCDACDABBABCDCABCDDCDABABCDACAACBDBBDCACCADBDDBABCDAACCABDBBDCACCADDBBDSemigroups〈σ2,σ11〉≅〈σ5,σ12〉, 〈σ2,σ12〉≅〈σ5,σ11〉, 〈σ2,σ13〉≅〈σ5,σ10〉, 〈σ3,σ6〉≅〈σ4,σ9〉, and 〈σ3,σ9〉≅〈σ4,σ6〉 have five elements. Semigroups 〈σ2,σ3〉≅〈σ4,σ5〉, 〈σ2,σ4〉≅〈σ3,σ5〉, and 〈σ2,σ5〉 have six elements. Construction of Cayley tables for these semigroups, as well as other semigroups, we leave it to the readers. One can prepare such tables removing from the Cayley table for 𝕄 some columns and rows. Then change Kuratowski operations onto letters of the alphabet. For example, 〈σ0,σ2,σ5〉 has the following Cayley table:
(17)0ABCDEF00ABCDEFAAACCEECBBDBFDDFCCECCEECDDDFFDDFEEECCEECFFDFFDDF
Preparing the Cayley table for 〈σ0,σ2,σ5〉 we put σ0=0,σ2=A,σ5=B,σ7=C,σ8=D,σ10=E, and σ13=F. This table immediately shows that semigroups 〈σ2,σ5〉 and 〈σ0,σ2,σ5〉 have exactly two automorphisms. These are identities and restrictions of 𝔸.Semigroup〈σ6,σ9〉 has eight elements. Semigroups 〈σ2,σ9〉≅〈σ5,σ6〉 and 〈σ3,σ8〉≅〈σ4,σ7〉 have nine elements. Semigroups 〈σ2,σ3,σ8〉≅〈σ4,σ5,σ7〉, 〈σ2,σ4,σ7〉≅〈σ3,σ5,σ8〉, and 〈σ2,σ5,σ6〉 have ten elements. Semigroups 〈σ2,σ3,σ5〉≅〈σ2,σ4,σ5〉 have eleven elements. In the end, the semigroup 〈σ3,σ4〉 has twelve elements.Thus, the semigroup〈σ3,σ4〉 includes fifty-seven semigroups, among which there are ten groups, fourteen monoids, and also forty-three semigroups which are not monoids. These semigroups consist of twenty-eight types of nonisomorphic semigroups, two non-isomorphism types of groups, two non-isomorphism types of monoids which are not groups, and twenty four non-isomorphism types of semigroups which are not monoids.


## 9. Viewing Semigroups Contained in𝕄

### 9.1. Descriptive Data on Semigroups which Are Contained in𝕄

There are one hundred eighteen, that is,118=2·57+4, semigroups which are contained in 𝕄. These are fifty-seven semigroups contained in 〈σ3,σ4〉, fifty-seven monoids formed by adding σ0 to a semigroup contained in 〈σ3,σ4〉, groups 〈σ0〉, 〈σ1〉, and monoids 〈σ1,σ6〉=𝕄1 and 〈σ1,σ2〉=𝕄.There are fifty-six types of nonisomorphic semigroups in𝕄. These are twenty-eight non-isomorphism types of semigroups contained in 〈σ3,σ4〉, twenty-six types of nonisomorphic monoids formed by adding σ0 to a semigroup contained in 〈σ3,σ4〉, and also 𝕄1 and 𝕄. Indeed, adding σ0 to a semigroup which is not a monoid we obtain a monoid. In this way, we get twenty-four types of nonisomorphic monoids. Adding σ0 to a monoid contained in 〈σ3,σ4〉 we get two new non-isomorphism types of monoids. But, adding σ0 to a group contained in 〈σ3,σ4〉 we get no new type of monoid, since we get a monoid isomorphic to 〈σ2,σ10〉 or 〈σ2,σ6〉. The other two types are 𝕄1 and 𝕄.


### 9.2. Semigroups which Are Not Monoids

Below we have reproduced, using the smallest number of generators and the dictionary order, a list of all 43 semigroups, which are included in the𝕄 as follows:(1)
〈σ2,σ3〉={σ2,σ3,σ6,σ7,σ10,σ11}.(2)
〈σ2,σ3,σ5〉={σ2,σ3,σ5,σ6,…,σ13}.(3)
〈σ2,σ3,σ8〉=〈σ2,σ3,σ9〉=〈σ2,σ3,σ12〉=〈σ2,σ3,σ13〉={σ2,σ3,σ6,σ7,…,σ13}.(4)
〈σ2,σ4〉={σ2,σ4,σ6,σ8,σ10,σ12}.(5)
〈σ2,σ4,σ5〉={σ2,σ4,σ5,…,σ13}.(6)
〈σ2,σ4,σ7〉=〈σ2,σ4,σ9〉=〈σ2,σ4,σ11〉=〈σ2,σ4,σ13〉={σ2,σ4,σ6,σ7…,σ13}.(7)
〈σ2,σ5〉={σ2,σ5,σ7,σ8,σ10,σ13}.(8)
〈σ2,σ5,σ6〉=〈σ2,σ5,σ9〉=〈σ2,σ5,σ11〉=〈σ2,σ5,σ12〉={σ2,σ5,σ6,…,σ13}.(9)
〈σ2,σ7〉={σ2,σ7,σ10}.(10)
〈σ2,σ8〉={σ2,σ8,σ10}.(11)
〈σ2,σ9〉=〈σ2,σ6,σ13〉=〈σ2,σ7,σ12〉=〈σ2,σ8,σ11〉=〈σ2,σ11,σ12〉=〈σ2,σ11,σ13〉  =  〈σ2,σ12,σ13〉={σ2,σ6,σ7,…,σ13}.(12)
〈σ2,σ11〉=〈σ2,σ6,σ7〉={σ2,σ6,σ7,σ10,σ11}.(13)
〈σ2,σ12〉=〈σ2,σ6,σ8〉={σ2,σ6,σ8,σ10,σ12}.(14)
〈σ2,σ13〉=〈σ2,σ7,σ8〉={σ2,σ7,σ8,σ10,σ13}.(15)
〈σ3〉={σ3,σ7,σ11}.(16)
〈σ3,σ4〉={σ2,σ3,…,σ13}.(17)
〈σ3,σ5〉={σ3,σ5,σ7,σ9,σ11,σ13}.(18)
〈σ3,σ5,σ6〉=〈σ3,σ5,σ8〉=〈σ3,σ5,σ10〉=〈σ3,σ5,σ12〉={σ3,σ5,σ6,…,σ13}.(19)
〈σ3,σ6〉=〈σ3,σ10〉={σ3,σ6,σ7,σ10,σ11}.(20)
〈σ3,σ8〉=〈σ3,σ12〉={σ3,σ6,σ7,…,σ13}.(21)
〈σ3,σ9〉=〈σ3,σ13〉={σ3,σ7,σ9,σ11,σ13}.(22)
〈σ4〉={σ4,σ8,σ12}.(23)
〈σ4,σ5〉={σ4,σ5,σ8,σ9,σ12,σ13}.(24)
〈σ4,σ5,σ6〉=〈σ4,σ5,σ7〉=〈σ4,σ5,σ10〉=〈σ4,σ5,σ11〉={σ4,σ5,…,σ13}.(25)
〈σ4,σ6〉=〈σ4,σ10〉={σ4,σ6,σ8,σ10,σ12}.(26)
〈σ4,σ7〉=〈σ4,σ11〉={σ4,σ6,σ7,…,σ13}.(27)
〈σ4,σ9〉=〈σ4,σ13〉={σ4,σ8,σ9,σ12,σ13}.(28)
〈σ5,σ6〉=〈σ5,σ7,σ12〉=〈σ5,σ8,σ11〉=〈σ5,σ9,σ10〉=〈σ5,σ10,σ11〉=〈σ5,σ10,σ12〉  =  〈σ5,σ11,σ12〉={σ5,σ6,…,σ13}.(29)
〈σ5,σ7〉={σ5,σ7,σ13}.(30)
〈σ5,σ8〉={σ5,σ8,σ13}.(31)
〈σ5,σ10〉=〈σ5,σ7,σ8〉={σ5,σ7,σ8,σ10,σ13}.(32)
〈σ5,σ11〉=〈σ5,σ7,σ9〉={σ5,σ7,σ9,σ11,σ13}.(33)
〈σ5,σ12〉=〈σ5,σ8,σ9〉={σ5,σ8,σ9,σ12,σ13}.(34)
〈σ6,σ7〉=〈σ6,σ11〉=〈σ10,σ11〉={σ6,σ7,σ10,σ11}.(35)
〈σ6,σ8〉=〈σ6,σ12〉=〈σ10,σ12〉={σ6,σ8,σ10,σ12}.(36)
〈σ6,σ9〉=〈σ6,σ13〉=〈σ7,σ12〉=〈σ8,σ11〉=〈σ9,σ10〉=〈σ11,σ12〉  =  〈σ6,σ7,σ8〉=〈σ7,σ8,σ9〉  =  〈σ10,σ11,σ13〉=〈σ10,σ12,σ13〉={σ6,σ7,…,σ13}.(37)
〈σ7,σ8〉={σ7,σ8,σ10,σ13}.(38)
〈σ7,σ9〉={σ7,σ9,σ11,σ13}.(39)
〈σ7,σ10〉={σ7,σ10}.(40)
〈σ7,σ13〉={σ7,σ13}.(41)
〈σ8,σ9〉={σ8,σ9,σ12,σ13}.(42)
〈σ8,σ10〉={σ8,σ10}.(43)
〈σ8,σ13〉={σ8,σ13}.


### 9.3. Isomorphism Types of Semigroups Contained in𝕄

Systematize the list of all isomorphism types of semigroups contained in the monoid𝕄. Isomorphisms, which are restrictions of the isomorphism 𝔸, will be regarded as self-evident, and therefore they will not be commented on.(i)
The monoid𝕄 contains 12 groups with 2 isomorphism types. These are 7 one-element groups and 5 two-element groups.(ii)
The monoid𝕄 contains 8 two-element monoids with 2 isomorphism types. These are σ0 added to 6 one-element groups and 〈σ2,σ10〉≅〈σ5,σ13〉. Also, it contains 4 two-element semigroups, not monoids, with 2 isomorphism types. These are 〈σ7,σ10〉≅〈σ8,σ13〉 and 〈σ8,σ10〉≅〈σ7,σ13〉.(iii)
The monoid𝕄 contains 12 three-element monoids with 5 isomorphism types. These are σ0 added to 4 two-element groups and also 〈σ0,σ2,σ10〉≅〈σ0,σ5,σ13〉, 〈σ0,σ7,σ10〉≅〈σ0,σ8,σ13〉, 〈σ0,σ8,σ10〉≅〈σ0,σ7,σ13〉, and 〈σ2,σ6〉≅〈σ5,σ9〉.(iv)
The monoid𝕄 contains 6 three-element semigroups, not monoids, with 3 isomorphism types. These are 〈σ2,σ7〉≅〈σ5,σ8〉, 〈σ2,σ8〉≅〈σ5,σ7〉, and 〈σ3〉≅〈σ4〉.(v)
The monoid𝕄 contains 8 four-element monoids, each contains σ0, with 4 isomorphism types. These are semigroups from two preceding items that can be substantially extended by σ0.(vi)
The monoid𝕄 contains 5 four-element semigroups, not monoids, with 3 isomorphism types. These are 〈σ6,σ7〉≅〈σ8,σ9〉, 〈σ6,σ8〉≅〈σ7,σ9〉 and 〈σ7,σ8〉. These semigroups extended by σ0 yield 5 monoids, all which consist of five elements, with 3 isomorphism types.(vii)
The monoid𝕄 contains 10 five-element semigroups—not monoids, with 5 isomorphism types. These are 〈σ2,σ11〉≅〈σ5,σ12〉, 〈σ2,σ12〉≅〈σ5,σ11〉, 〈σ2,σ13〉≅〈σ5,σ10〉, 〈σ3,σ6〉≅〈σ4,σ9〉, and 〈σ3,σ9〉≅〈σ4,σ6〉. These semigroups extended by σ0 yield 10 monoids, all of which consist of six elements, with 5 isomorphism types. We get 10 new isomorphism types, since the semigroups are distinguished by semigroups 〈σ3〉 and 〈σ4〉, and by nonisomorphic semigroups 〈σ6,σ7〉, 〈σ6,σ8〉, and 〈σ7,σ8〉.(viii)
The monoid𝕄 contains 5 six-element semigroups, not monoids, with 3 isomorphism types. These are 〈σ2,σ3〉≅〈σ4,σ5〉, 〈σ2,σ4〉≅〈σ3,σ5〉, and 〈σ2,σ5〉. These semigroups extended by σ0 yield 5 monoids, all of which consist of seven elements, with 3 isomorphism types. We get 6 new isomorphism types, since the semigroups are distinguished by non isomorphic semigroups 〈σ6,σ7〉, 〈σ6,σ8〉, and 〈σ7,σ8〉.(ix)
The monoid𝕄 contains no seven-element semigroup, not a monoid, no eight-element monoid and the only semigroup 〈σ6,σ9〉 with exactly eight elements and the only monoid 〈σ0,σ6,σ9〉 with exactly nine elements.(x)
The monoid𝕄 contains 4 nine-element semigroups, not monoids, with 2 isomorphism types. These are 〈σ2,σ9〉≅〈σ5,σ6〉 and 〈σ3,σ8〉≅〈σ4,σ7〉. The semigroup 〈σ2,σ9〉 does not contain a semigroup isomorphic to 〈σ3〉, hence it is not isomorphic to 〈σ3,σ8〉. These semigroups extended by σ0 yield 4 monoids, all of which consist of ten elements, with 2 isomorphism types.(xi)
The monoid𝕄 contains 6 ten-element semigroups, not monoids, with 4 isomorphism types. These are 〈σ1,σ6〉, 〈σ2,σ3,σ8〉≅〈σ4,σ5,σ6〉, 〈σ2,σ4,σ7〉≅〈σ3,σ5,σ6〉, and 〈σ2,σ5,σ6〉. These semigroups (except 〈σ1,σ6〉) extended by σ0 yield 5 monoids, all of which consist of ten elements, with 3 isomorphism types. We get 6 new isomorphism types, since the semigroups are distinguished by not isomorphic semigroups 〈σ2,σ3〉, 〈σ2,σ4〉, and 〈σ2,σ5〉.(xii)
The monoid𝕄 contains 2 isomorphic semigroups, not monoids, which consist of 11 elements, that is, 〈σ2,σ3,σ5〉≅〈σ2,σ4,σ5〉. These semigroups extended by σ0 yield 2 isomorphic monoids, which consist of 12 elements. The monoid 𝕄 contains no larger semigroup with the exception of itself.


## 9.1. Descriptive Data on Semigroups which Are Contained in𝕄

There are one hundred eighteen, that is,118=2·57+4, semigroups which are contained in 𝕄. These are fifty-seven semigroups contained in 〈σ3,σ4〉, fifty-seven monoids formed by adding σ0 to a semigroup contained in 〈σ3,σ4〉, groups 〈σ0〉, 〈σ1〉, and monoids 〈σ1,σ6〉=𝕄1 and 〈σ1,σ2〉=𝕄.There are fifty-six types of nonisomorphic semigroups in𝕄. These are twenty-eight non-isomorphism types of semigroups contained in 〈σ3,σ4〉, twenty-six types of nonisomorphic monoids formed by adding σ0 to a semigroup contained in 〈σ3,σ4〉, and also 𝕄1 and 𝕄. Indeed, adding σ0 to a semigroup which is not a monoid we obtain a monoid. In this way, we get twenty-four types of nonisomorphic monoids. Adding σ0 to a monoid contained in 〈σ3,σ4〉 we get two new non-isomorphism types of monoids. But, adding σ0 to a group contained in 〈σ3,σ4〉 we get no new type of monoid, since we get a monoid isomorphic to 〈σ2,σ10〉 or 〈σ2,σ6〉. The other two types are 𝕄1 and 𝕄.


## 9.2. Semigroups which Are Not Monoids

Below we have reproduced, using the smallest number of generators and the dictionary order, a list of all 43 semigroups, which are included in the𝕄 as follows:(1)
〈σ2,σ3〉={σ2,σ3,σ6,σ7,σ10,σ11}.(2)
〈σ2,σ3,σ5〉={σ2,σ3,σ5,σ6,…,σ13}.(3)
〈σ2,σ3,σ8〉=〈σ2,σ3,σ9〉=〈σ2,σ3,σ12〉=〈σ2,σ3,σ13〉={σ2,σ3,σ6,σ7,…,σ13}.(4)
〈σ2,σ4〉={σ2,σ4,σ6,σ8,σ10,σ12}.(5)
〈σ2,σ4,σ5〉={σ2,σ4,σ5,…,σ13}.(6)
〈σ2,σ4,σ7〉=〈σ2,σ4,σ9〉=〈σ2,σ4,σ11〉=〈σ2,σ4,σ13〉={σ2,σ4,σ6,σ7…,σ13}.(7)
〈σ2,σ5〉={σ2,σ5,σ7,σ8,σ10,σ13}.(8)
〈σ2,σ5,σ6〉=〈σ2,σ5,σ9〉=〈σ2,σ5,σ11〉=〈σ2,σ5,σ12〉={σ2,σ5,σ6,…,σ13}.(9)
〈σ2,σ7〉={σ2,σ7,σ10}.(10)
〈σ2,σ8〉={σ2,σ8,σ10}.(11)
〈σ2,σ9〉=〈σ2,σ6,σ13〉=〈σ2,σ7,σ12〉=〈σ2,σ8,σ11〉=〈σ2,σ11,σ12〉=〈σ2,σ11,σ13〉  =  〈σ2,σ12,σ13〉={σ2,σ6,σ7,…,σ13}.(12)
〈σ2,σ11〉=〈σ2,σ6,σ7〉={σ2,σ6,σ7,σ10,σ11}.(13)
〈σ2,σ12〉=〈σ2,σ6,σ8〉={σ2,σ6,σ8,σ10,σ12}.(14)
〈σ2,σ13〉=〈σ2,σ7,σ8〉={σ2,σ7,σ8,σ10,σ13}.(15)
〈σ3〉={σ3,σ7,σ11}.(16)
〈σ3,σ4〉={σ2,σ3,…,σ13}.(17)
〈σ3,σ5〉={σ3,σ5,σ7,σ9,σ11,σ13}.(18)
〈σ3,σ5,σ6〉=〈σ3,σ5,σ8〉=〈σ3,σ5,σ10〉=〈σ3,σ5,σ12〉={σ3,σ5,σ6,…,σ13}.(19)
〈σ3,σ6〉=〈σ3,σ10〉={σ3,σ6,σ7,σ10,σ11}.(20)
〈σ3,σ8〉=〈σ3,σ12〉={σ3,σ6,σ7,…,σ13}.(21)
〈σ3,σ9〉=〈σ3,σ13〉={σ3,σ7,σ9,σ11,σ13}.(22)
〈σ4〉={σ4,σ8,σ12}.(23)
〈σ4,σ5〉={σ4,σ5,σ8,σ9,σ12,σ13}.(24)
〈σ4,σ5,σ6〉=〈σ4,σ5,σ7〉=〈σ4,σ5,σ10〉=〈σ4,σ5,σ11〉={σ4,σ5,…,σ13}.(25)
〈σ4,σ6〉=〈σ4,σ10〉={σ4,σ6,σ8,σ10,σ12}.(26)
〈σ4,σ7〉=〈σ4,σ11〉={σ4,σ6,σ7,…,σ13}.(27)
〈σ4,σ9〉=〈σ4,σ13〉={σ4,σ8,σ9,σ12,σ13}.(28)
〈σ5,σ6〉=〈σ5,σ7,σ12〉=〈σ5,σ8,σ11〉=〈σ5,σ9,σ10〉=〈σ5,σ10,σ11〉=〈σ5,σ10,σ12〉  =  〈σ5,σ11,σ12〉={σ5,σ6,…,σ13}.(29)
〈σ5,σ7〉={σ5,σ7,σ13}.(30)
〈σ5,σ8〉={σ5,σ8,σ13}.(31)
〈σ5,σ10〉=〈σ5,σ7,σ8〉={σ5,σ7,σ8,σ10,σ13}.(32)
〈σ5,σ11〉=〈σ5,σ7,σ9〉={σ5,σ7,σ9,σ11,σ13}.(33)
〈σ5,σ12〉=〈σ5,σ8,σ9〉={σ5,σ8,σ9,σ12,σ13}.(34)
〈σ6,σ7〉=〈σ6,σ11〉=〈σ10,σ11〉={σ6,σ7,σ10,σ11}.(35)
〈σ6,σ8〉=〈σ6,σ12〉=〈σ10,σ12〉={σ6,σ8,σ10,σ12}.(36)
〈σ6,σ9〉=〈σ6,σ13〉=〈σ7,σ12〉=〈σ8,σ11〉=〈σ9,σ10〉=〈σ11,σ12〉  =  〈σ6,σ7,σ8〉=〈σ7,σ8,σ9〉  =  〈σ10,σ11,σ13〉=〈σ10,σ12,σ13〉={σ6,σ7,…,σ13}.(37)
〈σ7,σ8〉={σ7,σ8,σ10,σ13}.(38)
〈σ7,σ9〉={σ7,σ9,σ11,σ13}.(39)
〈σ7,σ10〉={σ7,σ10}.(40)
〈σ7,σ13〉={σ7,σ13}.(41)
〈σ8,σ9〉={σ8,σ9,σ12,σ13}.(42)
〈σ8,σ10〉={σ8,σ10}.(43)
〈σ8,σ13〉={σ8,σ13}.


## 9.3. Isomorphism Types of Semigroups Contained in𝕄

Systematize the list of all isomorphism types of semigroups contained in the monoid𝕄. Isomorphisms, which are restrictions of the isomorphism 𝔸, will be regarded as self-evident, and therefore they will not be commented on.(i)
The monoid𝕄 contains 12 groups with 2 isomorphism types. These are 7 one-element groups and 5 two-element groups.(ii)
The monoid𝕄 contains 8 two-element monoids with 2 isomorphism types. These are σ0 added to 6 one-element groups and 〈σ2,σ10〉≅〈σ5,σ13〉. Also, it contains 4 two-element semigroups, not monoids, with 2 isomorphism types. These are 〈σ7,σ10〉≅〈σ8,σ13〉 and 〈σ8,σ10〉≅〈σ7,σ13〉.(iii)
The monoid𝕄 contains 12 three-element monoids with 5 isomorphism types. These are σ0 added to 4 two-element groups and also 〈σ0,σ2,σ10〉≅〈σ0,σ5,σ13〉, 〈σ0,σ7,σ10〉≅〈σ0,σ8,σ13〉, 〈σ0,σ8,σ10〉≅〈σ0,σ7,σ13〉, and 〈σ2,σ6〉≅〈σ5,σ9〉.(iv)
The monoid𝕄 contains 6 three-element semigroups, not monoids, with 3 isomorphism types. These are 〈σ2,σ7〉≅〈σ5,σ8〉, 〈σ2,σ8〉≅〈σ5,σ7〉, and 〈σ3〉≅〈σ4〉.(v)
The monoid𝕄 contains 8 four-element monoids, each contains σ0, with 4 isomorphism types. These are semigroups from two preceding items that can be substantially extended by σ0.(vi)
The monoid𝕄 contains 5 four-element semigroups, not monoids, with 3 isomorphism types. These are 〈σ6,σ7〉≅〈σ8,σ9〉, 〈σ6,σ8〉≅〈σ7,σ9〉 and 〈σ7,σ8〉. These semigroups extended by σ0 yield 5 monoids, all which consist of five elements, with 3 isomorphism types.(vii)
The monoid𝕄 contains 10 five-element semigroups—not monoids, with 5 isomorphism types. These are 〈σ2,σ11〉≅〈σ5,σ12〉, 〈σ2,σ12〉≅〈σ5,σ11〉, 〈σ2,σ13〉≅〈σ5,σ10〉, 〈σ3,σ6〉≅〈σ4,σ9〉, and 〈σ3,σ9〉≅〈σ4,σ6〉. These semigroups extended by σ0 yield 10 monoids, all of which consist of six elements, with 5 isomorphism types. We get 10 new isomorphism types, since the semigroups are distinguished by semigroups 〈σ3〉 and 〈σ4〉, and by nonisomorphic semigroups 〈σ6,σ7〉, 〈σ6,σ8〉, and 〈σ7,σ8〉.(viii)
The monoid𝕄 contains 5 six-element semigroups, not monoids, with 3 isomorphism types. These are 〈σ2,σ3〉≅〈σ4,σ5〉, 〈σ2,σ4〉≅〈σ3,σ5〉, and 〈σ2,σ5〉. These semigroups extended by σ0 yield 5 monoids, all of which consist of seven elements, with 3 isomorphism types. We get 6 new isomorphism types, since the semigroups are distinguished by non isomorphic semigroups 〈σ6,σ7〉, 〈σ6,σ8〉, and 〈σ7,σ8〉.(ix)
The monoid𝕄 contains no seven-element semigroup, not a monoid, no eight-element monoid and the only semigroup 〈σ6,σ9〉 with exactly eight elements and the only monoid 〈σ0,σ6,σ9〉 with exactly nine elements.(x)
The monoid𝕄 contains 4 nine-element semigroups, not monoids, with 2 isomorphism types. These are 〈σ2,σ9〉≅〈σ5,σ6〉 and 〈σ3,σ8〉≅〈σ4,σ7〉. The semigroup 〈σ2,σ9〉 does not contain a semigroup isomorphic to 〈σ3〉, hence it is not isomorphic to 〈σ3,σ8〉. These semigroups extended by σ0 yield 4 monoids, all of which consist of ten elements, with 2 isomorphism types.(xi)
The monoid𝕄 contains 6 ten-element semigroups, not monoids, with 4 isomorphism types. These are 〈σ1,σ6〉, 〈σ2,σ3,σ8〉≅〈σ4,σ5,σ6〉, 〈σ2,σ4,σ7〉≅〈σ3,σ5,σ6〉, and 〈σ2,σ5,σ6〉. These semigroups (except 〈σ1,σ6〉) extended by σ0 yield 5 monoids, all of which consist of ten elements, with 3 isomorphism types. We get 6 new isomorphism types, since the semigroups are distinguished by not isomorphic semigroups 〈σ2,σ3〉, 〈σ2,σ4〉, and 〈σ2,σ5〉.(xii)
The monoid𝕄 contains 2 isomorphic semigroups, not monoids, which consist of 11 elements, that is, 〈σ2,σ3,σ5〉≅〈σ2,σ4,σ5〉. These semigroups extended by σ0 yield 2 isomorphic monoids, which consist of 12 elements. The monoid 𝕄 contains no larger semigroup with the exception of itself.


## 10. Cancellation Rules Motivated by Some Topological Properties

### 10.1. Some Consequences of the Axiom∅=∅-

So far, we used only the following relations (above named cancellation rules):σ2∘σ2=σ2, σ1∘σ1=σ0, σ2∘σ12=σ6, and σ2∘σ13=σ7. When one assumes X≠∅=∅-, then
(18)X=σ0(X)=σ2(X)=σ5(X)=σ7(X)=σ8(X)=σ10(X)=σ13(X),∅=σ1(X)=σ3(X)=σ4(X)=σ6(X)=σ9(X)=σ11(X)=σ12(X).
Using the substitution A↦Ac, one obtains equivalent relations between operations from the set {σ1,σ3,σ4,σ6,σ9,σ11,σ12}, and conversely. Therefore, cancellation rules are topologically reasonable only between the operations from the monoid as follows:
(19)〈σ0,σ2,σ5〉={σ0,σ2,σ5,σ7,σ8,σ10,σ13}.
Chapman, see [3], considered properties of subsets with respect to such relations. Below, we are going to identify relations that are determined by some topological spaces; compare [10, 11].


### 10.2. The Relationσ0=σ2

If a topological spaceX is discrete, then there exist two Kuratowski operation, only. These are σ0 and σ1. So, the monoid of Kuratowski operations reduced to the group 〈σ1〉.The relationσ0=σ2 is equivalent to any relation σ0=σi, where i∈{5,7,8,10,13}. Any such relation implies that every subset of X has to be closed and open; that is, X has to be discrete. However, one can check these using (only) the facts that σ1 is an involution and σ2 is an idempotent and the cancellation rules, that is, the Cayley table for 𝕄. So, σ0=σ2 follows
(20)σ0=σ2=σ5=σ7=σ8=σ10=σ13.


### 10.3. The Relationσ2=σ5

Topologically,σ2=σ5 means that X must be discrete. This is so because Ac-c⊆A⊆A- for any A⊆X.


### 10.4. The Relationσ2=σ7

Topologically, the relationσ2=σ7 implies σ0=σ2. But it requires the use of topology axioms ∅=∅- and C-∪B-=(C∪B)- for each C and B.Lemma 5.
For any topological spaceσ2=σ7 implies σ2=σ8.Proof.
Ifσ2=σ7, then A≠∅⇒Ac-c≠∅, for any A⊆X. Indeed, if Ac-c=∅, then σ7(A)=∅-=∅. Since A≠∅, then σ2(A)≠∅. Hence σ2(A)≠σ7(A), a contradiction.
The axiomC-∪B-=(C∪B)- implies that always
(21)(A-∩A-c-)c-c=∅.
Thus, the additional assumption σ2=σ7 follows that always A-∩A-c-=∅. Therefore, always A-c-=A-c, but this means that any closed set has to be open; in other words, σ2=σ8.Proposition 6.
For any topological spaceσ2=σ7 implies σ0=σ2.Proof.
But the relationσ2=σ7 is equivalent with σ5=σ8. By Lemma 5, we get σ2=σ5. Finally σ0=σ2.The relationσ2=σ7 has interpretation without the axiom ∅=∅-. Indeed, suppose X={a,b}. Put
(22)σ2(∅)={a}=σ2({a}),X=σ2(X)=σ2({b}).
Then, check that σ2=σ7 and
(23)σ8(∅)=σ5({a})=σ4({b})=σ1(X)=∅;
in other words, σ2=σ7 and σ2≠σ8. However, σ2=σ7 is equivalent to σ5=σ8.This relation impliesσ2=σ7=σ10, σ5=σ8=σ13, σ3=σ6=σ11, and σ4=σ9=σ12. For this interpretation, the monoid 𝕄/(σ2=σ7), consisting of Kuratowski operation over a such X, has six elements, only. In 𝕄/(σ2=σ7), there are relations covered by the following proposition, only.Proposition 7.
For any monoid with the Cayley table as for𝕄, the relation σ2=σ7 implies (i)
σ2=σ7=σ10=σ7∘σ2;(ii)
σ5=σ1∘σ2∘σ1=σ1∘σ7∘σ1=σ8=σ5∘σ2=σ5∘σ7=σ13;(iii)
σ3=σ2∘σ1=σ7∘σ1=σ6=σ10∘σ1=σ11;(iv)
σ4=σ1∘σ2=σ1∘σ7=σ9=σ1∘σ10=σ12.Thus, the Cayley table does not contain the complete information resulting from the axioms of topology.


### 10.5. The Relationσ2=σ8

Topologically, the relationσ2=σ8 means that any closed set is open, too. Thus, if X={a,b,c} is a topological space with the open sets X, ∅, {a,b}, and {c}, then 𝕄/(σ2=σ8) is the monoid of all Kuratowski operations over X. Relations σ2=σ8 and σ5=σ7 are equivalent. They imply relations: σ2=σ8=σ10, σ5=σ7=σ13, σ3=σ9=σ11, and σ4=σ6=σ12. The permutation
(24)(σ0σ1σ2σ3σ4σ5σ0σ1σ5σ4σ3σ2)
determines the isomorphism between monoids 𝕄/(σ2=σ7) and 𝕄/(σ2=σ8).


### 10.6. The Relationsσ2=σ10 and σ2=σ13

Topologically, the relationσ2=σ8 means that any nonempty closed set has nonempty interior. For each A, the closed set A-∩A-c- has empty interior, so the relation σ2=σ10 implies that A-c is closed. Hence, any open set has to be closed, so it implies σ2=σ8. The relation σ2=σ13 follows that each closed set has to be open, so it implies σ2=σ8, too.


### 10.7. The Relationσ7=σ8

Using the Cayley table for𝕄, one can check that the relations σ7=σ8 and σ10=σ13 are equivalent. Each of them gives σ7=σ8=σ10=σ13 and σ6=σ9=σ11=σ12. If X={a,b} is a topological space with the open sets X, ∅, and {a}, then 𝕄/(σ7=σ8) is the monoid of all Kuratowski operations over X and consists of 8 elements.


### 10.8. The Relationσ7=σ10

Using the Cayley table for𝕄, one can check that the relations σ7=σ10, σ8=σ13, σ6=σ11, and σ9=σ12 are equivalent. If X is a sequence converging to the point g and g∈X, then 𝕄/(σ7=σ10) is the monoid of all Kuratowski operations over X and consists of 10 elements.


### 10.9. The Relationσ7=σ13

Using the Cayley table for𝕄, one can check that the relations σ7=σ13 and σ8=σ10 are equivalent. Also σ6=σ12 and σ9=σ11. These relations give the monoid with the following Cayley table, where the row and column marked by the identity are omitted:
(25)σ1σ2σ3σ4σ5σ6σ7σ8σ9σ1σ0σ4σ5σ2σ3σ8σ9σ6σ7σ2σ3σ2σ3σ6σ7σ6σ7σ8σ9σ3σ2σ6σ7σ2σ3σ8σ9σ6σ7σ4σ5σ4σ5σ8σ9σ8σ9σ6σ7σ5σ4σ8σ9σ4σ5σ6σ7σ8σ9σ6σ7σ6σ7σ8σ9σ8σ9σ6σ7σ7σ6σ8σ9σ6σ7σ6σ7σ8σ9σ8σ9σ8σ9σ6σ7σ6σ7σ8σ9σ9σ8σ6σ7σ8σ9σ8σ9σ6σ7If a spaceX is extremally disconnected, then the closures of open sets are open; compare [2, page 452]. It follows that σ6=σ12. The space X={a,b} with the open sets X, ∅, and {a} is extremally disconnected. But it contains a one-element open and dense set {a}, and it follows that σ7=σ8. Similar is for the space βN; see [2, pages 228 and 453] to find the definition and properties of βN. There are Hausdorff extremally disconnected spaces which are dense in itself. For example, the Stone space of the complete Boolean algebra of all regular closed subsets of the unit interval, compare [12]. For such spaces σ7≠σ8 and σ7=σ13. To see this, suppose a Hausdorff X is extremally disconnected and dense in itself. Let X=U∪V∪W, where sets U,V, and W are closed and open. Consider a set A=Ac-c∪B∪C, such that (i)
Cc-c=∅ and C-=W;(ii)
∅≠B⊆V and B-c-c=∅;(iii)
U=Ac-c-≠Ac-c. Then check that (i)
σ0(A)=A and σ1(A)=X∖(Ac-c∪B∪C);(ii)
σ2(A)=U∪B-∪W and σ3(A)=X∖Ac-c;(iii)
σ4(A)=V∖B- and σ5(A)=Ac-c;(iv)
σ6(A)=σ12(A)=V and σ7(A)=σ13(A)=U;(v)
σ8(A)=σ10(A)=U∪W and σ9(A)=σ11(A)=V∪W. Hence we have that σ7≠σ8. Note that, if W=∅, then σ7(A)=σ8(A). This is the case of subsets of βN.


## 10.1. Some Consequences of the Axiom∅=∅-

So far, we used only the following relations (above named cancellation rules):σ2∘σ2=σ2, σ1∘σ1=σ0, σ2∘σ12=σ6, and σ2∘σ13=σ7. When one assumes X≠∅=∅-, then
(18)X=σ0(X)=σ2(X)=σ5(X)=σ7(X)=σ8(X)=σ10(X)=σ13(X),∅=σ1(X)=σ3(X)=σ4(X)=σ6(X)=σ9(X)=σ11(X)=σ12(X).
Using the substitution A↦Ac, one obtains equivalent relations between operations from the set {σ1,σ3,σ4,σ6,σ9,σ11,σ12}, and conversely. Therefore, cancellation rules are topologically reasonable only between the operations from the monoid as follows:
(19)〈σ0,σ2,σ5〉={σ0,σ2,σ5,σ7,σ8,σ10,σ13}.
Chapman, see [3], considered properties of subsets with respect to such relations. Below, we are going to identify relations that are determined by some topological spaces; compare [10, 11].


## 10.2. The Relationσ0=σ2

If a topological spaceX is discrete, then there exist two Kuratowski operation, only. These are σ0 and σ1. So, the monoid of Kuratowski operations reduced to the group 〈σ1〉.The relationσ0=σ2 is equivalent to any relation σ0=σi, where i∈{5,7,8,10,13}. Any such relation implies that every subset of X has to be closed and open; that is, X has to be discrete. However, one can check these using (only) the facts that σ1 is an involution and σ2 is an idempotent and the cancellation rules, that is, the Cayley table for 𝕄. So, σ0=σ2 follows
(20)σ0=σ2=σ5=σ7=σ8=σ10=σ13.


## 10.3. The Relationσ2=σ5

Topologically,σ2=σ5 means that X must be discrete. This is so because Ac-c⊆A⊆A- for any A⊆X.


## 10.4. The Relationσ2=σ7

Topologically, the relationσ2=σ7 implies σ0=σ2. But it requires the use of topology axioms ∅=∅- and C-∪B-=(C∪B)- for each C and B.Lemma 5.
For any topological spaceσ2=σ7 implies σ2=σ8.Proof.
Ifσ2=σ7, then A≠∅⇒Ac-c≠∅, for any A⊆X. Indeed, if Ac-c=∅, then σ7(A)=∅-=∅. Since A≠∅, then σ2(A)≠∅. Hence σ2(A)≠σ7(A), a contradiction.
The axiomC-∪B-=(C∪B)- implies that always
(21)(A-∩A-c-)c-c=∅.
Thus, the additional assumption σ2=σ7 follows that always A-∩A-c-=∅. Therefore, always A-c-=A-c, but this means that any closed set has to be open; in other words, σ2=σ8.Proposition 6.
For any topological spaceσ2=σ7 implies σ0=σ2.Proof.
But the relationσ2=σ7 is equivalent with σ5=σ8. By Lemma 5, we get σ2=σ5. Finally σ0=σ2.The relationσ2=σ7 has interpretation without the axiom ∅=∅-. Indeed, suppose X={a,b}. Put
(22)σ2(∅)={a}=σ2({a}),X=σ2(X)=σ2({b}).
Then, check that σ2=σ7 and
(23)σ8(∅)=σ5({a})=σ4({b})=σ1(X)=∅;
in other words, σ2=σ7 and σ2≠σ8. However, σ2=σ7 is equivalent to σ5=σ8.This relation impliesσ2=σ7=σ10, σ5=σ8=σ13, σ3=σ6=σ11, and σ4=σ9=σ12. For this interpretation, the monoid 𝕄/(σ2=σ7), consisting of Kuratowski operation over a such X, has six elements, only. In 𝕄/(σ2=σ7), there are relations covered by the following proposition, only.Proposition 7.
For any monoid with the Cayley table as for𝕄, the relation σ2=σ7 implies (i)
σ2=σ7=σ10=σ7∘σ2;(ii)
σ5=σ1∘σ2∘σ1=σ1∘σ7∘σ1=σ8=σ5∘σ2=σ5∘σ7=σ13;(iii)
σ3=σ2∘σ1=σ7∘σ1=σ6=σ10∘σ1=σ11;(iv)
σ4=σ1∘σ2=σ1∘σ7=σ9=σ1∘σ10=σ12.Thus, the Cayley table does not contain the complete information resulting from the axioms of topology.


## 10.5. The Relationσ2=σ8

Topologically, the relationσ2=σ8 means that any closed set is open, too. Thus, if X={a,b,c} is a topological space with the open sets X, ∅, {a,b}, and {c}, then 𝕄/(σ2=σ8) is the monoid of all Kuratowski operations over X. Relations σ2=σ8 and σ5=σ7 are equivalent. They imply relations: σ2=σ8=σ10, σ5=σ7=σ13, σ3=σ9=σ11, and σ4=σ6=σ12. The permutation
(24)(σ0σ1σ2σ3σ4σ5σ0σ1σ5σ4σ3σ2)
determines the isomorphism between monoids 𝕄/(σ2=σ7) and 𝕄/(σ2=σ8).


## 10.6. The Relationsσ2=σ10 and σ2=σ13

Topologically, the relationσ2=σ8 means that any nonempty closed set has nonempty interior. For each A, the closed set A-∩A-c- has empty interior, so the relation σ2=σ10 implies that A-c is closed. Hence, any open set has to be closed, so it implies σ2=σ8. The relation σ2=σ13 follows that each closed set has to be open, so it implies σ2=σ8, too.


## 10.7. The Relationσ7=σ8

Using the Cayley table for𝕄, one can check that the relations σ7=σ8 and σ10=σ13 are equivalent. Each of them gives σ7=σ8=σ10=σ13 and σ6=σ9=σ11=σ12. If X={a,b} is a topological space with the open sets X, ∅, and {a}, then 𝕄/(σ7=σ8) is the monoid of all Kuratowski operations over X and consists of 8 elements.


## 10.8. The Relationσ7=σ10

Using the Cayley table for𝕄, one can check that the relations σ7=σ10, σ8=σ13, σ6=σ11, and σ9=σ12 are equivalent. If X is a sequence converging to the point g and g∈X, then 𝕄/(σ7=σ10) is the monoid of all Kuratowski operations over X and consists of 10 elements.


## 10.9. The Relationσ7=σ13

Using the Cayley table for𝕄, one can check that the relations σ7=σ13 and σ8=σ10 are equivalent. Also σ6=σ12 and σ9=σ11. These relations give the monoid with the following Cayley table, where the row and column marked by the identity are omitted:
(25)σ1σ2σ3σ4σ5σ6σ7σ8σ9σ1σ0σ4σ5σ2σ3σ8σ9σ6σ7σ2σ3σ2σ3σ6σ7σ6σ7σ8σ9σ3σ2σ6σ7σ2σ3σ8σ9σ6σ7σ4σ5σ4σ5σ8σ9σ8σ9σ6σ7σ5σ4σ8σ9σ4σ5σ6σ7σ8σ9σ6σ7σ6σ7σ8σ9σ8σ9σ6σ7σ7σ6σ8σ9σ6σ7σ6σ7σ8σ9σ8σ9σ8σ9σ6σ7σ6σ7σ8σ9σ9σ8σ6σ7σ8σ9σ8σ9σ6σ7If a spaceX is extremally disconnected, then the closures of open sets are open; compare [2, page 452]. It follows that σ6=σ12. The space X={a,b} with the open sets X, ∅, and {a} is extremally disconnected. But it contains a one-element open and dense set {a}, and it follows that σ7=σ8. Similar is for the space βN; see [2, pages 228 and 453] to find the definition and properties of βN. There are Hausdorff extremally disconnected spaces which are dense in itself. For example, the Stone space of the complete Boolean algebra of all regular closed subsets of the unit interval, compare [12]. For such spaces σ7≠σ8 and σ7=σ13. To see this, suppose a Hausdorff X is extremally disconnected and dense in itself. Let X=U∪V∪W, where sets U,V, and W are closed and open. Consider a set A=Ac-c∪B∪C, such that (i)
Cc-c=∅ and C-=W;(ii)
∅≠B⊆V and B-c-c=∅;(iii)
U=Ac-c-≠Ac-c. Then check that (i)
σ0(A)=A and σ1(A)=X∖(Ac-c∪B∪C);(ii)
σ2(A)=U∪B-∪W and σ3(A)=X∖Ac-c;(iii)
σ4(A)=V∖B- and σ5(A)=Ac-c;(iv)
σ6(A)=σ12(A)=V and σ7(A)=σ13(A)=U;(v)
σ8(A)=σ10(A)=U∪W and σ9(A)=σ11(A)=V∪W. Hence we have that σ7≠σ8. Note that, if W=∅, then σ7(A)=σ8(A). This is the case of subsets of βN.



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*Source: 289854-2013-03-06.xml*