# Monadic Effect Algebras

Authors: Yuxi Zou; Xiaolong Xin
Journal: Journal of Mathematics
 (2022)

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## Abstract

The main goal of this paper is to introduce and investigate the related theory on monadic effect algebras. First, we design the axiomatic system of existential quantifiers on effect algebras and then use it to give the definition of the universal quantifier and monadic effect algebras. Then, we introduce relatively complete subalgebra and prove that there exists a one-to-one correspondence between the set of all the existential quantifiers and the set of all the relatively complete subalgebras. Moreover, we characterize and give the generated formula of monadic ideals and prove that Riesz monadic ideals and Riesz monadic congruences can be mutually induced. Finally, we study the strong existential quantifier and characterize monadic simple and monadic subdirectly irreducible effect algebras.


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## Body

## 1. Introduction

Effect algebras have been introduced by Foulis and Bennett [1] for modeling unsharp measurement in the quantum mechanical system. They are a generalization of many structures which arise in the axiomatization of quantum mechanics (Hilbert space effects), noncommutative measure theory and probability (orthomodular lattices and posets), fuzzy measure theory, and many-valued logic (MV-algebras). Moreover, several attempts have been made to study ideals and congruences on effect algebras, staring points are in the paper [2, 3] in which it is proved that there exists an order isomorphism between Riesz ideals and Riesz congruences and developed in [4, 5].The name “monadic” comes from the connection with predicate logics for languages having one placed predicate and a single quantifier. Monadic Boolean algebras are Boolean algebras with an additional unary operation which is an algebraic counterpart of the existential quantifier; they were introduced by Halmos [6] as an algebraic counterpart of the one-variable fragment of the classical predicate logic. Algebraic counterparts of the existential or universal quantifiers have also been consequently studied for certain nonclassical logics. For example, monadic MV-algebras were introduced and studied in [7], which is an algebraic model of the one-element fragment of Łukasiewicz predicate logic. Also, the theory of monadic MV-algebras has been developed in [8, 9]. Then, monadic noncommutative GMV-algebras having the same sense for noncommutative Łukasiewicz logic were introduced and studied in [10]. Moreover, monadic Heyting algebras were introduced in [11] as an algebraic model of the one-variable fragment of the intuitionistic predicate logic. From then on, the notion of monadic has been extended to other logical algebras such as monadic BL-algebras [12–14], monadic residuated lattices [15], monadic R0-algebras [16], monadic hoop algebras [17], monadic equality algebras [18], monadic quantum B-algebras [19, 20], and their noncommutative cases. Note that, since both MV-algebras and GMV-algebras satisfy De Morgan and double negation laws, in the definition of the corresponding monadic algebras, it is impossible to use only one of the existential and universal quantifiers as initial; the other is then definable as the dual of the original one. However, in contrast to the monadic MV-algebras and GMV-algebras, monadic Heyting algebras, monadic BL-algebras, and other logical algebras require using both kinds of quantifiers simultaneously because these quantifiers are not mutually definable.Originally, effect algebras were introduced as partial algebraic structure, and several attempts have been made to combine effect algebras with total algebraic structure [21–23]. However, previously, “monadic” was just introduced and studied on total algebraic structure. In our paper, we try to introduce the “monadic” into the effect algebra which is a typical partial algebraic structure. We believe that this can be helpful to study the effect algebras better.The paper is organized as follows. In Section2, we recall some basic definitions and properties in effect algebras which will be used in the remainder of the paper. In Section 3, we design the axiomatic system of existential quantifiers on effect algebras. Since effect algebras also satisfy the double negation laws, we just use existential quantifier to define monadic effect algebras. Also, the universal quantifier is definable as the dual of the existential quantifiers. Then, we give some nontrivial examples for monadic effect algebras and study some properties of existential quantifiers and universal quantifiers. Moreover, we introduce relatively complete subalgebras of effect algebras and study the relationship between the existential quantifiers and the relatively complete subalgebras. In Section 4, we introduce monadic ideals on monadic effect algebras. In particular, we focus on the relationship between Riesz monadic ideals and Riesz monadic congruences and the relationship between the lattice of all monadic ideals of E,∃ and the lattice of all ideals of ∃E. Moreover, we consider in which case the quotient structure E/I,∃I is also a monadic effect algebra. Finally, we introduce and characterize monadic simple and monadic subdirectly irreducible effect algebras.


## 2. Preliminaries

In this section, we summarize some definitions and results about effect algebras, which will be used in the following part, and we shall not cite them every time they are used.Definition 1 (see [1]).
An effect algebra (EA for short) is a structureE;⊕,0,1 consisting of a set E, elements 0 and 1 in E are the zero and the unit, and a partially defined operation ⊕ on E is called the orthosummation, such that for all x,y,z∈E:EA1: if x⊕y is defined, then y⊕x is defined and x⊕y=y⊕xEA2: if x⊕y and x⊕y⊕z are defined, then y⊕z and x⊕y⊕z are defined and x⊕y⊕z=x⊕y⊕zEA3: for each x∈E, there exists a unique element x¬∈E called the orthosupplement of x, such that x⊕¬x=1EA4:x⊕1 is defined only if x=0Remark 1 (see [1]).
In an effect algebraE, a partial ordering is defined by a≤b iff there is c∈E such that a⊕c=b. It turns out that the element c, if it exists, is uniquely defined. If a⊕b is defined, we say that a and b are orthogonal and write a⊥b. Moreover, a⊥b if and only if a≤¬b. Moreover, if a∨b and a∧b exist for all a,b∈E, then we say that E is a lattice ordered effect algebra.
The partial operation⊖ by b⊖a:=c iff a⊕c=b. Clearly, z is defined iff a≤b. With respect to this partial order, we have 0≤a≤1 for all a∈E. In particular, a¬=1⊖a. Moreover, if a⊥b, then a⊕b=a¬⊖b¬=b¬⊖a¬. The other partial operation ⊙ by a⊙b:=a¬⊕b¬¬. Clearly, a⊙b is defined iff a¬≤b iff b¬≤a.Proposition 1 (see [1]).
LetE be an effect algebra and a,b,c∈E. Then,(1)
Ifa≤¬b, then a≤a⊕b and a⊕b⊖a=b(2)
Ifa≤¬b≤¬c, then a⊕b⊖c=a⊕b⊖c(3)
Ifa≤b≤c, then a⊕c⊖b=c⊖b⊖a and also c⊖b⊕b⊖a=c⊖aDefinition 2 (see [1]).
LetE be an effect algebra and F⊆E. F is called a subalgebra of E if (i) 0,1∈F, (ii) for p∈F, p¬∈F, and (iii) for p,q∈F with p⊥q, p⊕q∈F.Definition 3 (see [3]).
A subsetI of an effect algebra E is called an ideal, if for all a,b∈E:I1: if a,b∈I and a⊥b, then a⊕b∈II2: if a∈I and b≤a, then b∈IAn idealI of an effect algebra E is called a Riesz ideal, if for all a,c,d∈E, where c⊥dI3: if a∈I and c⊕d≥a, then there exists h,k∈I with h≤c,k≤d and h⊕d≥a
An idealI of E is called a proper ideal if I≠E. Also, a proper ideal I of E is called a maximal ideal if it is not contained in any proper ideal of E. We denote the set of all ideals of E by IE and the set of all Riesz ideals of E by RIE. Moreover, for any X⊆E, there will be a smallest ideal X with X⊆X. We call X the ideal generated by X.Lemma 1 (see [3]).
LetE be an effect algebra and X⊆E. Then, the ideal X generated by X can be constructed in the following way: define I0=X, In+1:=∪x∈InE0,x, and In+2:=x⊕y:x,y∈In+1,x⊥y, n=0,1,⋯. Clearly, In⊆In+1⊆In+2, n=0,1,⋯. Then, X=∪n=0∞In.Definition 4 (see [3]).
A congruence on an effect algebraE is an equivalence relation ∼ such that for all a,a1,b,b1∈E:C1: ifa1∼a, b1∼b, a⊥b, and a1⊥b1, then a1⊕b1∼a⊕bC2: ifa1∼a and a⊥b, then there exists b0∈E with a1⊥b0 and b0∼bA congruence∼ on E is a Riesz congruence provided that the following holds for all a,b∈EC3: ifa∼b, then there exists c∈E such that c⊥a, c⊥b, and a⊕c∼1∼b⊕c
Given an idealI of an effect algebra E, we may define a relation ∼I on E as follows: a∼Ib⇔there exist i,j∈I: i≤a, j≤ba⊖i≤b, and b⊖j≤a. According to the relationship between ideals and congruences on effect algebras, we have the following theorem.Theorem 1 (see [3]).
LetE be an effect algebra. Then, the following holds:(i)
IfI is a Riesz ideal, then ∼I is a Riesz congruence and 0∼I=I(ii)
If∼ is a Riesz congruence, then I=0∼ is a Riesz ideal and ∼I=∼(iii)
The map which assigns to every Riesz congruence the equivalence class of 0 is an order isomorphism between the lattice of all Riesz congruence and the lattice of all Riesz ideal, whose inverse is the mapI↦∼ITheorem 2 (see [3]).
For a congruence∼ on an effect algebra E, the set of all congruence class is denoted by E/∼, i.e., E/∼=x|x∈E. We define a⊥b if there exists a1,b1∈E such that a1∼a, b1∼b, and a1⊥b1 and put a⊕b=a1⊕b1. Then, E/∼,⊕,0,1 is an effect algebra which is called a quotient effect algebra of E.Definition 5 (see [4]).
LetE;⊕,0,1 be an effect algebra. Then,(1)
E is called simple, if it has exactly two ideals: 0 and E(2)
E is called subdirectly irreducible if among the nontrivial congruence relations of E, there exists the least oneDefinition 6.
(see [8]). Let M=M,⊕,⊙,∗,0,1 be an MV-algebra. A mapping ∃:M⟶M is called an existential quantifier on M, if for all x,y∈M, the following conditions hold:E1x≤∃xE2∃x∨y=∃x∨∃yE3∃∃x∗=∃x∗E4∃∃x⊕∃y=∃x⊕∃yE5∃x⊙y=∃x⊙∃yE6∃x⊕y=∃x⊕∃y


## 3. Monadic Effect Algebra

In this section, we use the existential quantifier to construct the axiomatic system of monadic effect algebras. Also, we introduced relatively complete subalgebras of effect algebras and prove that there exists a one-to-one correspondence between the set of the existential quantifiers and the set of relatively complete subalgebras.Definition 7.
LetE=E,⊕,0,1 be an effect algebra. A mapping ∃:E⟶E is called an existential quantifier on E, if for all x,y∈E, the following conditions hold:E1:x≤∃xE2: if x≤y, then ∃x≤∃yE3∃∃¬x=∃¬xE4∃∃x⊕∃y=∃x⊕∃y, whenever ∃x⊕∃y is definedExample 1.
LetE=E,⊕,0,1 is an effect algebra. Clearly, we have that idE is an existential quantifier on E.
From the above example, we know that monadic effect algebra is a generalization of effect algebra.Example 2.
LetE=0,1. For any x,y∈E, we define x⊕y=x+y if and only if x+y≤1 and x¬=1−x, then E,⊕,0,1 is an effect algebra. For any n∈N, we define ∃n as follows:(1)∃nx=0,x=0,in,x∈i−1n,ini=1,2,…,n.
We can check that∃n is an existential quantifier on E.Example 3.
LetE=0,a,b,c,d,1, where 0≤a,b≤c,d≤1. Define operation ⊕ and ¬ as follows (see Tables and 21):
Then, (E,⊕, ¬, 0, 1) is an effect algebra. We define ∃ as follows: ∃0=0,∃a=∃d=d,∃b=b,and∃c=∃1=1. We can easily check that ∃ is an existential quantifier on E.Table 1⊕0abcd100abcd1aadc1——bbc——1—cc1————dd—1———11—————Table 2¬01acbdcadb10Remark 2.
As is well known, we call a lattice ordered effect algebra which satisfiesa∧b=0⇒a≤b¬ as an MV-effect algebra, and MV-algebras and MV-effect algebras are in one-to-one correspondence; we always identify them. However, the existential quantifiers on MV-algebras and MV-effect algebras are different. We can check that an existential quantifier on MV-algebra is an existential quantifier on MV-effect algebra, but the converse not holds. Let E,∃ be the monadic effect algebra given by Example 3; we can check that E is an MV-effect algebra. Since a⊕a=b and ∃a=d, ∃a⊕∃a=d⊕d is not defined, which implies that ∃ does not satisfy the E6 in the definition of monadic MV-algebras.Proposition 2.
LetM be an MV-algebra and ∃ be an existential quantifier on E. Then, for all x,y∈E, the following conditions are valid:(1)
∃1=1 and ∃0=0(2)
∃∃x=∃x(3)
x≤∃y⇔∃x≤∃y(4)
∃¬x≤∃¬x(5)
If∃x⊕∃y is defined, then ∃x⊕y≤∃x⊕∃yx(6)
The set∃E is an effect subalgebra, ∃E=x∈E:∃x=x, and ∃ on ∃E is the identity on ∃E(7)
Ify≤x, then ∃∃x⊖∃y=∃x⊖∃y(8)
Ify≤x, then ∃x⊖∃y≤∃x⊖y(9)
∃∃x⊙∃y=∃x⊙∃y, whenever ∃x⊙∃y is defined(10)
Ifx¬≤y, then ∃x⊙y≤∃x⊙∃y(11)
∃∃x∧∃y=∃x∧∃y, whenever ∃x∧∃y existProof.
(1)
ByE1, 1≤∃1; thus, ∃1=1 as 1 is the greatest element of E. By E3, ∃0=∃1¬=∃∃¬1=∃¬1=1¬=0.(2)
By (1) andE4, ∃∃x=∃∃x⊕0=∃∃x⊕∃0=∃x⊕∃0=∃x⊕0=∃x.(3)
Ifx≤∃y, by E2 and (2), we have ∃x≤∃∃y=∃y; conversely, if ∃x≤∃y, by (1), we have x≤∃x≤∃y.(4)
By (1), we havex¬≤∃x¬; then, ∃¬x≤x¬≤∃x¬.(5)
If∃x⊕∃y is defined, then we have ∃x≤¬∃y, so x≤∃x≤¬∃y≤¬y; then, we get x⊕y defined. Therefore, ∃x⊕y≤∃∃x⊕∃y=∃x⊕∃y.(6)
For anyx∈∃E, there exists y∈E, such that x=∃y; then, we have ∃x=∃∃y=∃y=x. Therefore, ∃E=x∈E:∃x=x. For any x,y∈∃E, we have x=∃x and y=∃y; if x⊕y is defined, then x⊕y=∃x⊕∃y=∃∃x⊕∃y. Therefore, x⊕y∈∃E. For any x∈∃E, then x=∃x, so ∃¬x=∃¬∃x=¬∃x=¬x. Therefore, x¬∈∃E. By the above, we have ∃E as an effect subalgebra. Finally, since ∃∃x=∃x, we have ∃ on ∃E as the identity on ∃E.(7)
Ify≤x, then ∃y≤∃x, so ∃x⊖∃y is defined. Now, we assume ∃x⊖∃y=z; then, ∃x⊖∃y=∃¬x⊕∃y¬=∃∃¬x⊕∃y¬=∃∃∃¬x⊕∃y¬=∃∃∃∃¬x⊕∃y¬, so z∈∃E. Then, we have z=∃z. Therefore, ∃∃x⊖∃y=∃z=z=∃x⊖∃y.(8)
Ify≤x, then there exists z∈E, such that y⊕z=x, so ∃x⊖∃y=∃y⊕z⊖∃y≤∃y⊕∃z⊖∃y=∃z=∃x⊖y.(9)
∃∃x⊙∃y=∃∃¬x⊕∃¬y¬=∃∃∃¬x⊕∃∃¬y¬=∃∃¬∃∃¬x⊕∃∃¬y=∃¬∃∃¬x⊕∃∃¬y=∃¬∃∃¬x⊕∃∃¬y=∃¬x⊕∃¬y¬=∃x⊙∃y.(10)
Ifx¬≤y is defined, then x¬≤y, so ∃¬x≤∃y, and so ∃¬x≤∃x¬≤∃y. Then, we have ∃x⊙∃y defined. Therefore, ∃x⊙y≤∃∃x⊙∃y=∃x⊙∃y.(11)
Assume thatd=∃x∧∃y is defined in E. We show that ∃d=d. Check, ∃d≤∃∃x=∃x and ∃d≤∃∃y=∃y. This yields ∃d≤∃x∧∃y=d, so ∃d=d. Then, ∃∃x∧∃y=∃x∧∃y.
In monadic MV-algebras, the existential quantifier∃ and the universal quantifier ∀ can be mutually induced. Similarly, by means of ∃, we define ∀ on E by the rule(2)∀x:=¬∃¬xR.Proposition 3.
LetE be an effect algebra and ∃ be an existential quantifier on E. If ∀ is defined by R, then the following conditions are satisfied:A1:∀x≤xA2: if x≤y, then ∀x≤∀yA3∀∀¬x=∀¬xA4:∀∀x⊙∀y=∀x⊙∀y, whenever ∀x⊙∀y is definedProof.
A1: since x¬≤∃x¬, then x=¬¬x≥¬∃¬x=∀xA2: if x≤y, then y¬≤x¬, so ∃¬y≤∃¬x, and then ∀x=∃x¬¬≤∃y¬¬=∀yA3∀∀¬x=∃¬∀¬¬x=∃¬∃x¬¬=y¬¬∃x¬=∀¬xA4∀∀x⊙∀y=∃∃x¬¬⊙∃y¬¬¬¬=∃¬∃x¬⊕∃y¬=∃x¬⊕∃y¬¬=∃x¬¬¬⊕∃y¬¬¬¬=∀x⊙∀y
An unary operation∀:E⟶E on an effect algebra that satisfies A1−A4 will be called a universal quantifier.Proposition 4.
LetE=E,⊕,0,1 be an effect algebra and ∀ be a universal quantifier on E. Define ∃x=¬∀¬x; then, ∃ is an existential quantifier on E.Proof.
It is similar to the above proposition.
From Propositions2 and 3, we get that for an effect algebra, there exists a one-to-one correspondence between existential quantifiers and universal quantifiers. Therefore, alike monadic MV-algebras, we can choose only one between the existential quantifiers and the universal quantifiers to construct the axiomatic system of monadic effect algebras. In order to study the monadic ideals and the related theory on effect algebra and for an ideal I on an effect algebra, when ∃x∈I, we can get ∀x∈I, but the reverse side not holds. Therefore, we use the existential quantifiers to construct the axiomatic system of monadic effect algebras which is more natural and convenient.Definition 8.
LetE be an effect algebra and ∃ be an existential quantifier on E. Then, the couple E,∃ is called a monadic effect algebra.Proposition 5.
Let∀ be a universal quantifier on an effect algebra E. For all x,y∈E, the following conditions are valid:(1)
∀0=0 and ∀1=1(2)
∀∀x=∀x(3)
∀x≤y⇔∀x≤∀y(4)
∀¬x≤¬∀x(5)
If∀x⊙∀y is defined, then ∀x⊙∀y≤∀x⊙y(6)
∀∀x⊕∀y=∀x⊕∀y, whenever ∀x⊕∀y is defined(7)
Ifx⊕y is defined, then ∀x⊕∀y≤∀x⊕y(8)
The set∀E is an effect subalgebra, ∀E=x∈E:∀x=x, and ∀ on ∀E is the identity on ∀E(9)
Ify≤x, then ∀∀x⊖∀y=∀x⊖∀y(10)
Ify≤x, then ∀x⊖y≤∀x⊖∀y(11)
∀∀x∨∀y=∀x∨∀y, whenever ∀x∨∀y existsProof.
(1) Since∀0≤0, we have ∀0=0 as 0 is the least element of E. ∀1=∀¬0=¬¬∀¬0=¬∃0=¬0=1.(2)∀∀x=∀∀x⊙1=∀∀x⊙∀1=∀x⊙∀1=∀x⊙1=∀x.(3) If∀x≤y, we have ∀x≤∀∀y=∀y; conversely, if ∀x≤∀y, we have ∀x≤∀y≤y.(4) Since∀x≤x, we have ∀¬x≤¬x≤¬∀x.(5) If∀x⊙∀y is defined, then we have ∀¬x≤∀y, so x¬≤∀¬x≤∀y≤y; therefore, x⊙y is defined. So, ∀x⊙∀y=∀∀x⊙∀y≤∀x⊙y.(6)∀∀x⊕∀y=∀∀¬x⊙∀¬y¬=∀∀∀¬x⊙∀∀¬y¬=∀∀¬∀∀¬x⊙∀∀¬y=∀¬∀∀¬x⊙∀∀¬y=∀∀¬x⊙∀∀¬y¬=∀¬x⊙∀¬y¬=∀x⊕∀y.(7) Ifx⊕y is defined, then x≤¬y; thus, ∀x≤x≤¬y≤¬∀y. Therefore, ∀x⊕∀y is defined. So, ∀x⊕∀y=∀∀x⊕∀y≤∀x⊕y.(8 and 9) Similar to (6) and (7) of Proposition2.(10) Ify≤x, then there exists z such that y⊕z=x and ∀y≤∀x, so ∀x⊖∀y=∀y⊕z⊖∀y≥∀y⊕∀z⊖∀y=∀z=∀x⊖y.(11) Assume thatd=∀x∨∀y is defined in E. We show that ∀d=d. Check ∀d≥∀∀x=∀x and ∀d≥∀∀y=∀y. This yields ∀d≥∀x∨∀y=d, so ∀d=d. Then, ∀∀x∨∀y=∀x∨∀y.Proposition 6.
LetE,∃ be a monadic effect algebra and ∀ be the universal quantifier defined by R. For all x,y∈E, the following conditions are valid:(1)
∀∃x=∃x and ∃∀x=∀x(2)
∀x=x⇔∃x=x(3)
∃¬x=∀x¬ and ∀¬x=∃x¬(4)
∀∃x⊙∃y=∃x⊙∃y and ∃∀x⊙∀y=∀x⊙∀y, whenever ∃x⊙∃y and ∀x⊙∀y are defined(5)
∀∃x⊕∃y=∃x⊕∃y and ∃∀x⊕∀y=∀x⊕∀y, whenever ∃x⊙∃y and ∀x⊙∀y are defined(6)
Ify≤x, then ∃∀x⊖∀y=∀x⊖∀y and ∀∃x⊖∃y=∃x⊖∃y(7)
∃,∀ establishes a Galois connectionProof.
(1)
∀∃x=¬∃¬∃x=¬¬∃x=∃x and ∃∀x=¬∀¬∀x=¬¬∀x=∀x.(2)
If∀x=x, then we have x∈∀E, so there exists y, such that ∀y=x. Thus, ∃x=∃∀y=∀y=x. Conversely, if ∃x=x, then we have x∈∃E, so there exists y, such that ∃y=x. Thus, ∀x=∀∃y=∃y=x.(3)
x¬∃x=∀¬¬x¬=∀x¬ and ∀¬x=∃¬¬x¬=∃x¬.(4)
∀∃x⊙∃y=∀∀∃x⊙∀∃y=∀∃x⊙∀∃y=∃x⊙∃y. ∃∀x⊙∀y=∃∃∀x⊙∃∀y=∃∀x⊙∃∀y=∀x⊙∀y.(5)
∀∃x⊕∃y=∀∀∃x⊕∀∃y=∀∃x⊕∀∃y=∃x⊕∃y. ∃∀x⊕∀y=∃∃∀x⊕∃∀y=∃∀x⊕∃∀y=∀x⊕∀y.(6)
Ify≤x1, then ∀x⊖∀y and ∃x⊖∃y are defined. ∃∀x⊖∀y=∃∃∀x⊖∃∀y=∃∀x⊖∃∀y=∀x⊖∀y and ∀∃x⊖∃y=∀∀∃x⊖∀∃y=∀∃x⊖∀∃y=∃x⊖∃y.(7)
∃∀x=∀x≤x and ∀∃x=∃x≥x, so ∃,∀ establishes a Galois connection.
Let∀ be a universal quantifier on E. We denote the kernel of ∀ by Ker∀=x∈E:∀x=0. Moreover, we call ∀ as faithful if Ker∀=0.Proposition 7.
Let∀ be a universal quantifier on an effect algebra E. Then, the following conditions are valid:(1)
If∀ is faithful, then x<y implies ∀x<∀y(2)
∀ is faithful ⇔∀x=x, for any x∈E(3)
∀ is faithful ⇔∃x=x, for any x∈EProof.
(1)
Suppose thatx<y, and ∀x=∀y and ∀y⊖x≤∀y⊖∀x=0 giving y⊖x=0 so that x=y; this is a contradiction.(2)
For anyx∈E, ∀x⊖∀x≤∀x⊖∀∀x=0; since ∀ is faithful, we have x⊖∀x=0; then, x=∀x. The converse is clear.(3)
∀x=x⇔∃x=x.
Next, we study the relationship between the existential quantifiers and the relatively complete subalgebras on monadic effect algebras.
A subalgebraE0 of an effect algebra E is said to be relatively complete if for every a∈E, the set b∈E0|a≤b has the least element in E0, which is denoted by(3)infb∈E0|a≤bor∧ba≤b∈E0.Theorem 3.
LetE,∃ be a monadic effect algebra. Then, ∃E is a relatively complete subalgebra of E and ∃a=infb∈∃E|a≤b.Proof.
By Proposition2 (6), ∃E is a subalgebra of E. Let a∈E and A=b∈∃E|a≤b. Then, ∃a∈∃E and for any b∈A, we have b=∃b. So, ∃a≤∃b=b; thus, ∃a is the least element of A. Therefore, ∃E is a relatively complete subalgebra of E.Remark 3.
LetE be the effect algebra given in Example 2. Then, by Theorem 3, we know that, for any n∈N, the subset 0,1/n,2/n,…,n−1/n,1 is a relatively complete subalgebra of E.Theorem 4.
LetE=E;⊕,0,1 be an effect algebra and E0 be a relatively complete subalgebra of E. For any a∈E, define ∃a=infb∈E0|a≤b. Then, E=E;⊕,∃,0,1 is a monadic effect algebra.Proof.
We only need to prove∃ defined above which is an existential quantifier on E.E1: a≤b, where b∈E0|a≤b. Then, we get a≤infb∈E0|a≤b=∃a.E2: if x≤y, then b∈E0|y≤b⊆b∈E0|x≤b. So, infb∈E0|x≤b≤infb∈E0|y≤b, i.e., ∃x≤∃y.For anyx∈E0, since x≤x, we have infb∈E0|x≤b≤x, i.e., ∃x≤x. Also, by the definition of ∃, we have x≤∃x. Therefore, for any x∈E0, we have x=∃x.E3: for any x∈E, we have ∃x∈E0; since E0 is a subalgebra of E, we get ∃¬x∈E0, and hence ∃∃¬x=∃¬x.E4: ∃x,∃y∈E0, and E0 is a subalgebra of E. Hence, if ∃x⊕∃y is defined, then ∃x⊕∃y∈E0, and so we have ∃∃x⊕∃y=∃x⊕∃y.
Therefore,E=E;⊕,∃,0,1 is a monadic effect algebra.
LetAandB be two posets. Consider a function h:A⟶B; the function k:B⟶A is called left adjoint to h, if kb≤a⇔b≤ha for any a∈A and b∈B.Theorem 5.
LetE be an effect algebra. Then, there exists a one-to-one correspondence between each pair of the following sets:(1)
The set of all the existential quantifiers onE, denoted by EE(2)
The set of all the relatively complete subalgebras onE, denoted by RCSE(3)
The set of all the pairsE,E0, where E0 is a subalgebra of E and the canonical embedding h:E0⟶E has left adjoint function, denoted by PE,E0Proof.
(i)
Consider the mappingf:EE⟶RCSE, i.e., f∃=∃E. For any ∃∈EE, according to Theorem 3, ∃E is a relatively complete subalgebra of E. Also, consider the mapping g:RCSE⟶EE, i.e., gE0=∃′. For any relatively complete subalgebra E0∈RCSE, according to Theorem 4,∃′a=infb∈E0|a≤b is an existential quantifier on E. So, for any a∈E, we have ∃a= inf b∈∃E|a≤b=∃′a. Therefore, g°f=idEE. On the other hand, for any E0∈RCSE, suppose gE0=∃′; then, f∃′=∃′E. For any a∈E0, by the definition of ∃′, we have a=∃′a∈∃′E, so E0⊆∃′E. Conversely, for any a∈∃′E, there exists b∈E, which satisfies ∃′b=a. Since ∃′ is an existential quantifier, then ∃′a=∃′∃′b=∃′b=a; therefore, a=∃′a=infb∈E0|a≤b∈E0, i.e., ∃′E⊆E0. Therefore, ∃′E=E0, which implies that f°g=idRCSE. According to the above, there exists a one-to-one correspondence between (1) and (2).(ii)
LetE0 be a relatively complete subalgebra of E. We define ∃ha=infb∈E0|a≤b, and h:E0⟶E is the canonical embedding. If ∃ha≤b, where a∈E and b∈E0, then b∈c∈E0|a≤c; hence, a≤b=hb. Conversely, if a≤hb, then a≤b, so b∈c∈E0|a≤c, and so ∃ha≤b. Hence, ∃ha≤b⇔a≤hb. Therefore, the canonical embedding h:E0⟶E has left adjoint function, and the left adjoint function of h is ∃h. On the other hand, for a given pair E,E0, where E0 is a subalgebra of E, the canonical embedding h:E0⟶E has left adjoint function. Now, we prove infb∈E0|a≤b exists. For any a∈E, ∃ha∈E0. For any x∈b∈E0|a≤b, we have a≤x=hx, so ∃ha≤x. Also, ∃ha≤∃ha; then, a≤h∃ha=∃ha, so we have a≤∃ha. Hence, ∃ha is the least element in b∈E0|a≤b. Therefore, infb∈E0|a≤b=∃ha. And so, there exists a one-to-one correspondence between (2) and (3).(iii)
By (i) and (ii), we know that there exists a one-to-one correspondence between (1) and (3).


## 4. Monadic Ideals on Monadic Effect Algebras

In this section, we introduce and investigate monadic ideals on monadic effect algebras. Also, in order to get some more and better results, we introduce strong existential quantifier. Moreover, we introduce and characterize simple and subdirectly irreducible monadic effect algebras.Definition 9.
LetE=E;⊕,∃,0,1 be a monadic effect algebra and I be an ideal of E. Then, I is called a monadic ideal if for any i∈I, we have ∃i∈I.
WhenI is a Riesz ideal and I is also a monadic ideal, we say I is a Riesz monadic ideal. We denote the sets of all the monadic ideals and the Riesz monadic ideals of E,∃ by MIE and RMIE, respectively.Example 4.
For anyn∈N, let E,∃n be the monadic effect algebras given by Example 2. Then, 0 is the only monadic ideal in each E,∃n.Example 5.
LetE,∃ be the monadic effect algebra given in Example 3. Take I1=0,a,d and I2=0,b; we can check that I1 and I2 are monadic ideal of E,∃. I3=0,a is an ideal but not a monadic ideal. a∈I3, but ∃a=d∈I3.Example 6.
LetE,∃ be the monadic effect algebra given in Example 3. Take I1=0,a,d, I2=0,b, I3=0,a, and I4=0,c. We can check that I2 is a Riesz monadic ideal. I1 is not a Riesz monadic ideal as I1 is not a Riesz ideal. Since d≤a⊕c, h,k∈I1 does not exist such that d≤h⊕k. I3 is not a Riesz monadic ideal as I3 is not a monadic ideal. I4 is not a Riesz monadic ideal as I4 is neither a monadic ideal nor a Riesz ideal.Proposition 8.
LetE,∃ be a monadic effect algebra and X⊆E. The monadic ideal generated by X is denoted by X∃; then, X∃ can be constructed in the following way: define I0=X, In+1:=∪x∈∃InE0,x, and In+2:=x⊕y:x,y∈In+1,x⊥y, n=0,1,⋯. Clearly, In⊆In+1⊆In+2, n=0,1,⋯. Then, X∃=∪n=0∞In.Proof.
Clearly,X∃ is an ideal of E,∃. For any t∈X∃, there exists n∈N such that t∈In. If In=∪x∈∃In−1E0,x, then there exists y∈∃In−1 such that t≤y; then, ∃t≤∃y=y, which implies ∃t∈In⊆X∃. If In=x⊕y:x,y∈In−1,x⊥y, then there exist a,b∈In−1 such that t=a⊕b; then, ∃t=∃a⊕b and ∃a⊕b∈In+1, which implies ∃t∈In+1⊆X∃. Therefore, X∃ is a monadic ideal of E,∃.Definition 10.
LetE,∃ be a monadic effect algebra. If for any x,y∈E, x⊥y and x⊕y≠1 imply ∃x⊥∃y; then, we say the existential quantifier ∃ is a strong and E,∃ is a strong monadic effect algebra.Example 7.
LetE=E,⊕,0,1 be an effect algebra. We have that idE is a strong existential quantifier on E.Example 8.
LetE,∃ be the monadic effect algebra given by Example 3. Then, we have a⊥a on E and ∃a=d, but d⊥d. Hence, the existential quantifier ∃ is not strong.Example 9.
LetE={0, a, b, c, 1}. Define operation ⊕ and ¬ as follows:(see Tables 3 and 4):
Then, (E,⊕, ¬, 0, 1) is an effect algebra. We define ∃ as follows: ∃0=0,∃a=∃b=b,and∃c=∃1=1. We can easily check that ∃ is a strong existential quantifier on E.Table 3⊕0abc100abc1aabc1—bbc1——cc1———11————Table 4¬01acbbca10Proposition 9.
LetE,∃ be a strong monadic effect algebra a∈∃E and I be a monadic ideal. If J=a∪I, then J is also a monadic ideal.Proof.
For anyx∈J, since the ⊕ is commutative and associative, then there exist n∈N and i∈I such that x≤na⊕i. If na⊕i=1, then ∃x≤∃na⊕i=∃1=1. Then, ∃x∈J. If na⊕i≠1, then ∃x≤∃na⊕i≤∃na⊕∃i≤n∃a⊕∃i=na⊕∃i as ∃ is strong. Since ∃i∈I, then na⊕∃i∈J; it follows that ∃x∈J. Therefore, J is a monadic ideal of E,∃.Theorem 6.
LetE,∃ be a strong monadic effect algebra and I be an ideal of E. Then,(1)
I is a monadic ideal of E,∃ if and only if I=I∩∃E(2)
There exists a lattice isomorphism between the lattice of all monadic ideals ofE,∃ and the lattice of all ideals of ∃EProof.
(1)
LetE,∃ be a monadic effect algebra. I⊆E is a monadic ideal. Let i∈I, then ∃i∈I and ∃i∈I∩∃E; since i≤∃i, we get i∈I∩∃E, so I⊆I∩∃E. Moreover, since I∩∃E⊆I, then we have I∩∃E⊆I⊆I. Therefore, we get I=I∩∃E.Conversely, letI=I∩∃E; if I=E, clearly, I is a monadic ideal. If I≠E, for I0 in I∩∃E, we can easily check that ∃I0⊆I∩∃E. For I1 in I∩∃E, for any x∈I1, then there exists a∈I0 such that x≤a; then, we have ∃x≤∃a=a∈I0, so ∃x∈I1; hence, ∃I1∈I∩∃E. For I2 in I∩∃E, for any x∈I2, there exists a,b∈I1 such that a⊥b and x=a⊕b; then, ∃x=∃a⊕b≤∃a⊕∃b as ∃ is strong and a⊕b≠1. Since ∃a,∃b∈I1, then ∃a⊕∃b∈I2. We have ∃x∈I3. Therefore, we get ∃I2⊆I∩∃E. The rest may be deduced by analogy. Therefore, for any In in I∩∃E, we have ∃In⊆I∩∃E. Then, we have ∃I∩∃E⊆I∩∃E, so I∩∃E is a monadic ideal of E,∃. Therefore, we get that if I=I∩∃E, then I is a monadic ideal of E,∃.(2)
First, we prove that, for any idealI of ∃E, we have I=I∩∃E.IfI=∃E, clearly, we have I=I∩∃E. If I is proper, it is clear that I⊆I∩∃E. Conversely, for I0 in I, we have I0=I; then, ∃I0=I. For I1 in I, for any x∈I1, there exists y∈I0, such that x≤y; then, ∃x≤∃y. Since ∃I0=I and I is an ideal, we have ∃x∈I; then, ∃I1⊆I. For I2 in I, for any x∈I2, there exists a,b∈I1 such that a⊥b and x=a⊕b. If a¬∈I1, then ∃¬a∈∃I1⊆I, and so ∃¬a∈I, Since a∈I1, then ∃a∈∃I1⊆I, so ∃a⊕¬∃a=1∈I, which is a contradictory to I is proper, so a¬∈I1. Therefore, a⊕b≠1. Then, ∃x=∃a⊕b≤∃a⊕∃b as ∃ is strong and a⊕b≠1. Since I is an ideal, then we have ∃a⊕∃b∈I; it follows that ∃x∈I and so ∃I2⊆I. The rest may be deduced by analogy. We have that, for any In in I, we have ∃In⊆I. Also, for any In in I, for any x∈In, ∃x∈∃In⊆I. If x¬∈In, ∃¬x∈∃In⊆I, so ∃¬x∈I, and so ∃x⊕¬∃x=1∈I, which is a contradictory to I that is proper, so a¬∈In. It follows that if I is proper, then we have I that is proper. For I0 in I, I0∩∃E=I⊆I. For I1 in I, for any x∈I1∩∃E, there exists a∈I0 such that x≤a; then, x=∃x≤∃a∈I, so x∈I. Therefore, I1∩∃E⊆I. For I2 in I, for any x∈I2∩∃E, there exists a,b∈I1 such that x=a⊕b, so x=∃x=∃a⊕b≤∃a⊕∃b as ∃ is strong and a⊕b≠1. Since a,b∈I1, we have ∃a,∃b∈I, and so x≤∃a⊕∃b∈I. Therefore, I2∩∃E⊆I. The rest may be deduced by analogy. For any In in I, we have In∩∃E⊆I. Therefore, for any ideal I of ∃E, we have I=I∩∃E.
Next, we prove that there exists a lattice isomorphism between the lattice of all monadic ideals ofE,∃ and the lattice of all ideals of ∃E. We have already known that ∃E is an effect algebra. Then, correspondence I⟶I maps any ideal I of ∃E to the monadic ideal I of E,∃. Moreover, I=I∩∃E. On the other hand, let J be a monadic ideal of E,∃, then J=J∩∃E, where J∩∃E is an ideal of ∃E. So, we have found the required 1−1 correspondence between the set of monadic ideals of E,∃ and the set of ideals of ∃E. Moreover, it is clear that I1⊆I2⇒I1⊆I2 and J1⊆J2⇒J1∩∃E⊆J2∩∃E. Therefore, there exists a lattice isomorphism between the lattice of all monadic ideals of E and the lattice of all ideals of E0.Definition 11.
LetE=E;⊕,∃,0,1 be a monadic effect algebra and ∼ be a congruence relation on E. ∼ is called a monadic congruence relation if a∼b implies that ∃a∼∃b, for any a,b∈E.Proposition 10.
IfI is a Riesz monadic ideal of a monadic effect algebra E,∃, we define a relation ∼I on E,∃ as follows: a∼bI⇔ there exist i,j∈I:i≤a,j≤b,a⊖i≤b, and b⊖j≤a; then, ∼I is a monadic congruence of E,∃.Proof.
By Theorem1, we have already known that if I is a Riesz ideal, then ∼I is a congruence of E. Now, we prove ∼I is a monadic congruence of E. If a∼Ib, then there exists i,j∈I such that i≤a,j≤b,a⊖i≤b, and b⊖j≤a. Then, ∃a⊖∃i≤∃a⊖i≤∃b and ∃b⊖∃j≤∃b⊖j≤∃a; since I is a Riesz monadic ideal, we have ∃i,∃j∈I. Then, ∃a∼I∃b. Therefore, ∼I is a monadic congruence of E,∃.Theorem 7.
LetE,∃ be a monadic effect algebra. Then, the following holds:(i)
IfI is a Riesz monadic ideal, then ∼I is a Riesz monadic congruence and 0∼I=I(ii)
If∼ is a Riesz monadic congruence, then I=0∼ is a Riesz monadic ideal and ∼I=∼(iii)
There exists an order isomorphism between the lattice of all Riesz monadic congruence and the lattice of all Riesz monadic idealProof.
(a)
From Theorem 1 and Proposition 10, the result is stragightforward.(b)
By Theorem1, we get I=0∼ which is a Riesz ideal. So, we just prove I=0∼ is a Riesz monadic ideal. For any x∈0∼, we have x∼0. Since ∼ is a Riesz monadic congruence, then ∃x∼∃0=0, so ∃x∈0∼. Therefore, I=0∼ is a Riesz monadic ideal and ∼I=∼.(c)
From (a) and (b), we know that the mappingf:I↦∼I is a bijection, whose inverse is the map g:∼↦0∼. Now, we prove the mapping f and g are isotone. Let I and J be two Riesz monadic ideals of E,∃ such that I⊆J. If a∼Ib, then there exists i,j∈I such that a⊖i≤b and b⊖j≤a. Since i,j∈I⊆J, then a∼Jb, which means ∼I⊆∼J. On the other hand, let ∼ and ≈ be two Riesz monadic congruence of E,∃ such that ∼⊆≈. For any i∈0∼, we have i∼0; then, i≈0 as ∼⊆≈. It follows that i∈0≈, which means that 0∼⊆0≈. Hence, f and g are isotone. Therefore, f is the order isomorphism between the lattice of all Riesz monadic congruence and the lattice of all Riesz monadic ideal.
LetE,∃ be a monadic effect algebra and I be a Riesz monadic ideal. We define the mapping ∃I:E/I⟶E/I by ∃Ix=∃x for any x∈E.Proposition 11.
LetE,∃ be a strong monadic effect algebra and I be a Riesz monadic ideal of E,∃. Then, E/I;∃I is also a strong monadic effect algebra.Proof.
SinceI is a Riesz monadic ideal, then by Theorem 1, ∼I is a monadic congruence. We denote E/∼I=E/I. By Theorem 2, we have already known that E/I is an effect algebra. Now, we prove E/I;∃I is a strong monadic effect algebra.E1: since x≤∃x, then there exists y such that x⊥y and x⊕y=∃x. It follows that x⊥y and x⊕y=x⊕y=∃x. Therefore, x≤∃Ix.E2: if x≤y, then there exists a such that x⊥a and x⊕a=y. It follows that there exists x1,a1,y1∈E such that x1∼Ix,a1∼Ia,y1∼Iy, and x1⊕a1=y1, so x1≤y1. Then, ∃x1≤∃y1. There exists b∈E such that ∃x1⊕b=∃y1. Therefore, ∃Ix⊕b=∃x⊕b=∃x1⊕b=∃y1=∃Iy. Hence, ∃Ix≤∃Iy.E3∃I¬∃Ix=∃∃¬x=∃¬x=∃¬Ix.E4: let ∃Ix⊕∃Iy be defined. If ∃Ix⊕∃Iy=1, clearly, ∃I∃Ix⊕∃Iy=∃Ix⊕∃Iy. Otherwise, since ∃Ix⊥∃Iy, then there exists a,b∈E such that ∃x∼Ia, ∃y∼Ib, a⊥b, and ∃Ix⊕∃Iy=a⊕b. As ∃ is strong and a⊕b≠1, we have ∃a⊥∃b. Since ∃x∼aI, then ∃x∼∃Ia; it follows that a∼∃Ia. Similarly, we get b∼∃Ib. As ∼I is a monadic congruence, we have a⊕b∼∃Ia⊕∃b. Therefore, ∃I∃Ix⊕∃Iy=∃I∃x⊕∃y=∃Ia⊕b=∃I∃a⊕∃b=∃∃a⊕∃b=∃a⊕∃b=∃Ia⊕∃Ib=∃Ix⊕∃Iy.
Therefore,E/I;∃I is a monadic effect algebra. Moreover, if x⊥y and x⊕y≠1, then there exists a,b∈E such that x∼aI, y∼bI, and a⊥b. Then, ∃x∼∃Ia, ∃y∼∃Ib, and ∃a⊥∃b as ∃ is strong and a⊕b≠1. So, ∃x⊥∃y, which implies that ∃Ix⊥∃Iy. Therefore, E/I;∃I is a strong monadic effect algebra.Definition 12.
LetE,∃ be a monadic effect algebra. A proper monadic ideal I of E,∃ is called a maximal monadic ideal if it is not strictly contained in any proper monadic ideal of E,∃.Example 10.
LetE={0, a, b, c, d, e, f, 1}, where 0 ≤ a, b ≤ c, d ≤ 1. Define operation ⊕ and ¬ as follows (see Tables 5 and 6):
Then, (E,⊕, ¬, 0, 1) is an effect algebra. We define ∃ as follows: ∃0=0,∃a=∃d=d,∃b=b,and∃c=∃e=∃f=∃1=1. We can easily check that ∃ is an existential quantifier on E. Define I1=0,a,d,e; I1 is a maximal ideal, but not a maximal monadic ideal because ∃e=1∉I1. Define I2=0,b, and we can check if I2 is a maximal monadic ideal of E,∃.Table 5⊕0abcdef100abcdef1aadc1————bbc——1———cc1——————dd—1—————ee—————1—ff————1——11———————Table 6¬01acbdcadbeffe10Proposition 12.
LetI be a monadic ideal of a monadic effect algebra E,∃. If for any x∈E, we have ∃x∈I or ∃¬x∈I; then, I is a maximal monadic ideal of E,∃.Proof.
Suppose for anyx∈E, we have ∃x∈I or ∃¬x∈I. Assume I is not a maximal monadic ideal of E,∃. Then, there exists a proper monadic ideal I strictly containing I, so there also exists a∈J, but a∈I. Hence, ∃a∈I; according to our assumption, ∃a∈I. Therefore, ∃a∈I. On the other hand, as a∈J and J is a monadic ideal, we have ∃a∈J, so ∃¬a⊕∃a=1∈J. This contradicts the fact that J is a proper monadic ideal. Therefore, I is a maximal monadic ideal of E,∃.Definition 13.
A monadic effect algebraE,∃ is called monadic simple if it has exactly two monadic ideals: 0 and E.Example 11.
LetE,∃ be the monadic effect algebra given by Example 9. We can easily check that E,∃ is monadic simple.Example 12.
For anyn∈N, let E,∃n be the monadic effect algebra given by Example 2. We can check E,∃n is a simple monadic effect algebra.Proof.
0 is the only monadic ideal in each E,∃n.Theorem 8.
LetE,∃ be a strong monadic effect algebra. Then, the following conditions are equivalent:(1)
E,∃ is monadic simple(2)
∃E is simpleProof.
1⇒2: let I be an ideal of ∃E and I≠0. We will show that I=∃E. Consider J=x∈E|x≤i, for some i∈I. If x,y∈J, then there exists i,j∈I such that x⊕y≤i⊕j∈I, so x⊕y∈J. If y≤x and x∈J, then there exists i∈I such that y≤x≤i, so y∈J. Moreover, if x∈J, then there exists i∈I such that x≤i, so ∃x≤∃i=i and ∃x∈J. Therefore, J is a monadic ideal of E,∃. Since I⊆J, I≠0, and E,∃ is monadic simple, we have J=E, so 1∈J; by the definition of J, we have 1∈I. Therefore, we have I=∃E which implies ∃E is simple.2⇒1: let I be a monadic ideal of E,∃. Then, we have I∩∃E as an ideal of ∃E; since ∃E is simple, we get I1∩∃E=0, or I∩∃E=∃E. If I∩∃E=∃E, then ∃E⊆I, so 1∈I. Then, we deduce that I=E. If I∩∃E=0, take x∈I. ∃x∈I∩∃E=0, so for any x∈I, we have ∃x=0, but x≤∃x, so for any x∈I, we have x=0, so I=0. Therefore, E,∃ is monadic simple.Remark 4.
WhenE,∃ is a strong monadic effect algebra, there exists a more concise proof for the above theorem. Since ∃ is strong, by Theorem 6, we know there exists a lattice isomorphism between the lattice of all monadic ideals of E,∃ and the lattice of all ideals of ∃E, which implies that the number of the elements in the sets MIE and I∃E is equal. Therefore, when ∃ is strong, E,∃ is monadic simple if and only if ∃E is simple.
The result of Theorem8 shows that we can use the partial structure ∃E to characterize the total structure E. Also, ∃E has the same structure as E,∃, which reveals the significance of the image set ∃E.Definition 14.
A monadic effect algebraE,∃ is said to be monadic subdirectly irreducible if for all of the nontrivial monadic congruence relations of E,∃, there exists the least one.
LetE,∃ be monadic subdirectly irreducible effect algebra and ∼ be the least nontrivial monadic congruence of E,∃. Then, by Theorem 6, there is a Riesz monadic ideal I of E,∃ such that ∼I=∼, which means I is the least Riesz monadic ideal of E,∃ such that I≠0. Thus, we can conclude that E,∃ is monadic subdirectly irreducible if among the nontrivial Riesz monadic ideals of E,∃, there exists the least one.Proposition 13.
LetE,∃ be a subdirectly irreducible monadic effect algebra and I1,I2∈RMIE. If I1∩I2=0, then I1=0 or I2=0.Proof.
SupposeI1≠0 and I2≠0, then I1 and I2∈I∈RMIE|I≠0, so ∩I∈RMIE|I≠0⊆I1∩I2=0. Therefore, ∩I∈RMIE|I≠0=0, which contradicts to the fact that E,∃ is subdirectly irreducible. Therefore, I1=0 or I2=0.Proposition 14.
LetE,∃ be a monadic effect algebra. Then, the following conditions are equivalent:(1)
E,∃ is a subdirectly irreducible monadic effect algebra(2)
∩x∃|x∈E,x>0≠0, where x∃ is the monadic ideal generated by xProof.
1⇒2: let E,∃ be a subdirectly irreducible monadic effect algebra. We have ∩I∈RMIE|I≠0≠0; then, ∩x∃|x∈E,x>0≠0.2⇒1: let ∩x∃|x∈E,x>0≠0, then there exists a∈∩x∃|x∈E,x>0. Now, we prove that, for any I∈RMIE, if I≠0, we have a∈I. In fact, if I≠0, then there exist x∈Iandx>0; by assuming, we have a∈x∃, so a∈I. Hence, a∈∩I∈RMIE|I≠0. Therefore, ∩I∈RMIE|I≠0≠0, which implies E,∃ is a subdirectly irreducible monadic effect algebra.Theorem 9.
LetE,∃ be a monadic effect algebra and ∀ be faithful. Then, the following conditions are equivalent:(1)
E,∃ is monadic subdirectly irreducible(2)
∃E is a subdirectly irreducible subalgebra of EProof.
1⇒2: let E,∃be a subdirectly irreducible monadic effect algebra. Then, RMIE−0 has a minimal element I. We have already known ∃E is a subalgebra of E. Now, we will prove that I∩∃E is the minimal ideal of ∃E such that I∩∃E≠0. First, I∩∃E=0; since ∀I⊆I∩∃E=0, then we have ∀x=0 for any x∈I. Therefore, I⊆ker∀=0, and so I=0, which is a contradiction. Hence, I∩∃E≠0. Next, we show that I∩∃E is the minimal ideal of ∃E. Assume that J is an ideal of ∃E, so, clearly, J is a monadic ideal of E,∃ generated by J. By the minimality of I, we know that I⊆J, so I∩∃E⊆J∩∃E=J. Therefore, I∩∃E is the minimal ideal of ∃E such that I∩∃E≠0. Therefore, ∃E is a subdirectly irreducible subalgebra of E.2⇒1: let ∃E be a subdirectly irreducible subalgebra of E. Then, there is a minimal ideal I of ∃E such that I≠0. So, I is a monadic ideal of E,∃ such that I≠0. Now, we prove that I is the minimal monadic ideal of E,∃. Assume J is another nontrivial monadic ideal of E,∃; then, we have J∩∃E which is an ideal of ∃E. By the minimality of I, we have I⊆J∩∃E, so I⊆J∩∃E=J. Hence, I is the minimal monadic ideal of E,∃. Therefore, E,∃ is a subdirectly irreducible monadic effect algebra.


## 5. Conclusions

As effect algebras are partial algebraic structures, froma⊥b, we cannot get ∃a⊥∃b (naturally holds in total algebraic structure), which caused some difficulties for us to prove some propositions on monadic effect algebras. So, we introduce strong existential quantifiers as follows: if for any x,y∈E, x⊥y, and x⊕y≠1, imply ∃x⊥∃y. Also, we require x⊕y≠1 because if for any x,y∈E, x⊥y implies ∃x⊥∃y, we can prove ∃=idE. Another difference between monadic effect algebras and some monadic total algebras is that the class of monadic effect algebras just forms a quasivariety, but the class of monadic MV-algebras, monadic BL-algebras, and monadic residuated lattices forms a variety, respectively. Moreover, effect algebra is a model of unsharp measurement in quantum mechanical system. Therefore, using the existential quantifiers and the universal quantifiers to study the unsharp measurement in quantum mechanical system is our next work.



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*Source: 6323524-2022-09-19.xml*