数理解析研究所講究録
第 1562 巻 2007 年 61-72

61

Logico-Algebraic Structures for Information Integration in the Brain
Marcin Jan Schroeder
Akita International University
Akita, Japan

mjs@aiu.ac.jp
Abstract In the study of the brain mechanisms responsible for consciousness, the most mysterious, and probably the most important for mathematicalmodeling is the phenomenal unity of conscious awareness. The author has proposed in his earlier work an approach to explain this unity in terms of integrationof information, where information is understood as identification of a variety,and Its integration as transformation of the selective manifestation of information into structural one. Also in the earlier study, the general mathematicalmodel of integration has been exemplified using the process of color discrimination, and a hypothetical interpretation of the unity of consciousness has beenpresented in terms of the irreducibility of the algebraic structures involved inmodeling of integration.
The present paper is devoted to the issue of the relationship between partiallyordered sets and transitive closure operators which seem the core concepts ofthe model. The main question is about the way how the closure operation
can be selected from the class of closure operators compatible with the partial
order. It has been shown that the structure of orthocomplementation (or moregenerally of strong orthogonality relation) build over the partial order gives aunique selection of the closure operator. Introducing a generalized form of logic
into the process of integration turns out to be equivalent to the selection of theunique closure operator compatible with the logical structure.
1. Introduction
The present paper is a continuation of the earlier work on a mathematicalmodel for information integration. [1] As before, the ultimate goal is to provide
a mathematical model of the brain mechanisms responsible for consciousness.
Since the most outstanding, if not defining characteristic of consciousness is
its phenomenal unity, such mechanisms must involve processes of information
integration, and this is the reason why they are in the center of our interest.
Before we can discuss information integration, it is necessary to recollecthow in this and our earlier studies information is understood, since $inform*$tion has diverse and frequently fallacious conceptualizations. Information canbe defined in the framework of the philosophically fertile theme of the “onemany” relationship as the identification of a variety.[2] The identification canbe understood as any unifying aspect of the variety, such as selection of theone out of many, or as unification of the many into one. The distinction of thetwo mentioned modes of identification gives forth two fundamental manifesta
tions of information, the selective and the structural. However, they are only
different manifestations of the uniform phenomenon being in an ever present

62
dual relationship. The selection of the one out of many requires some structuralcharacteristics distinguishing selected element, on the other hand the structure
unifying the many into a whole is a selection of the one out of many ways of
unification.
Integration of information is understood as a process of information transformation in which its selective manifestation is replaced by structural. Whatshould be emphasized here is the fact that integration of information cannot be
reduced to its accumulation or its quantitative increase, but must involve someform of qualitative change. It is natural to expect that the outcome of such a
process will be characterized in terms of the unity or wholeness of some variety.
This is why we can expect that it is information integration which is responsible
for the unity of conscious experience into which a large variety of multi-modal
perceptions is integrated.
The history of the analysis of this unity in modern psychology, as well as theaccount of the attempts to explain it, has been presented elsewhere.[1] For the
purpose of making the present paper self-contained, it will be sufficient to recall
that the only approach free from the “homunculus fallacy” arising in all attempts
to model consciousness without taking into account an essential transformationof information in the cognitive processes, was based on the interpretation of theunity of consciousness as a result of the quantum entanglement (coherence) of
the processing units in the brain. However, “the possibility that the totality of
microtubules (.. .]our brains may well take part in global quantum coherence-or at least that there is sufficient quantum entanglement between the states ofdifferent microtubules across the brain. . .” [3] considered by Roger Penrose as
an opportunity for involving the quantum mechanical description in the studyof brain mechanisms, seems as unrealistic now, as it has been fifteen years ago.
This is why in the earlier paper the idea of searching for quantum-type
coherence has been initiated as a promising direction of inquiry, but without
incorporating all formalism of quantum mechanics, which seems to be not suitable for description of the brain as a physical system. For someone familiar
only with the standard Hilbert space formalism of quantum mechanics in whichquantum coherence is simply superposition of wave functions, this idea may
seem as incomprehensible as an attempt to contemplate a smile in the absenceof the face. However, in a more abstract formalism of quantum mechanics, theso called quantum logic, quantum coherence has a very simple, yet fundamental algebraic interpretation in terms of the irreducibility of the lattice of closed
subsets of the Hilbert space, or more generally, of the lattice of quantum logical
propositions. $[1, 4]$Thus, we can explore the possibility that the unity of consciousness is a resultof irreducibility of the lattice, or if there is need for increased generality, of thepartially ordered set, of some basic elements which have equally fundamental
function in the description of brain mechanisms as the basic yes-no experiments
in quantum mechanics. Our original idea was based on the heuristic speculativeargument that the most likely structure of this type could be a complete latticeof all closed subsets of the set of neurons, or other functional units in the brain,with respect to some unidentified yet transitive closure operation. It has to
$\ln$

63
be stressed that the reference to closure operators and complete lattices of theclosed subsets is not necessary for such a model utilizing the concept of algebraic
irreducibility, as the irreducibility of partially ordered sets can be consideredinstead of the irreducibility of lattices. The heuristic argument for searchingamong closure operators for the formalism of information integration was theiromnipresence in mathematics (logic, topology, geometry, probability, etc.) andthe intuitive association of the process of uniting the variety of all subsets into
the restricted Moore family of closed subsets with the process of integration of
the variety of perceptions into the uniform objects of conscious awareness.
In the case of quantum mechanics, the lattice of closed subsets of the Hilbert
space (here too we have an instance of a closure operation,) has a function of
the empirical logic for physical characteristics of the system. However, it would
be an error to conclude that if a concept of logic is involved, we should follow the track of the calculus of logical operations and look for a fundamental
partial order structure in the computational models of the brain. The computational metaphor for the brain is the main source of the homunculi fallacies“populating” the domain of Artificial Intelligence. Also, the logic of computation is based on the Boolean algebra, which is an extreme case of completely
de-coherent structure, or more precisely of a structure which can be completely
reduced into the direct product of simple two element substructures. Thus, if wewant to take advantage of the uniting characteristic of the quantum coherence,
we should look for the formalism somewhere else.
In our earlier paper another partially ordered set has been considered. Its
simple instance is identified in the simplified model of integration of informationin the process of color vision which can be described as follows.
The model has the form of a Venn diagram for three sets (for that reason it
was called a Venn gate) with the two sets of arrows. The eight arrows on the left
side terminating in each of the eight regions of the Venn diagram represent thevariety of eight basic colors of the rainbow (including white and black). Each ofthem can activate appropriate receptors represented by the circles of the Venn
diagram, and these activations form the output marked by the three arrows on
the right side. The selection of one of the variety of eight colors is transformed
into structural configuration of activations in receptors. Each selection produces
unique pattern of activations, if the input light is homogeneous.
For instance, the yellow light is uniquely represented as the activation of two
receptors, with each of them representing another color (red and green respectively) when activated separately. However, when several different input lights
are coming at the same time, the pattern may be the same for a combination of
inputs as for a single input. For instance, the output pattern for the yellow color
can appear when the input consists of the first and third arrows corresponding
to the red and green light. The possibility of representing the yellow color as a
combination of the representations for green and red makes it a greater element
than the other two in the partial ordering induced on the input set.
What is critical for the understanding of the model (which otherwise would
be completely trivial account of the tri-color vision) is the fact, that what makes
the receptor a processing unit for the green color in perception is not its sensi-

64
tivity to the light of particular length, but the way how it is (wired’ into thebrain mechanisms. If the same receptor is re-wired in place of the receptor processing red light, we would perceive grass as red. There is no reason to believethat the brain simply “knows” to what light its receptors are sensitive. Thus,
color discrimination is not so much the matter of the chemistry of light sensitive substances, but of the internal organization of the brain and its peripheralssuch as the retina. Moreover, it suggests that the actual functional units in thecolor discrimination are not receptors, but some processing units (gates,) eachinvolving three receptors (possibly more) organized as in the model above.
In our particular elementary model of integration of information in the simplified process of color discrimination, the induced partial order is a Boolean
algebra, which is not of a great interest for us in our search for the source of theunity of conscious awareness. However, nothing prevents us from building models of similar “integrating gates” which induce quantum-like irreducible partial

orders.
The present paper reports further exploration of this idea with the focus on
the relationship between partial orders and transitive closure operators.
2. Transitive closure operators compatible with partial order
The simple model of integration of information involved in color discrimination can be easily generalized when we observe that the “processing gate”
is essentially a function from the (unstructured) set of inputs to the structure
built on the outputs, in our particular example a Boolean algebra . Eachoutput itself is a subset of the set of atoms of the lattice on which this algebra
is built, where an atom in a partially ordered set with the least element is
an element greater than , but not greater than any other element. The partial order induced on the inputs is generated by the inclusion of representing
them subsets of atoms. We have here something which could be interpreted as a“logarithmic” set operation, as opposed to constructing “power sets”. Thus, thequestion is whether the structure which we want to extract from the process andutilize for modeling information integration is a partial order, partial order withorthocomplementation (Boolean algebra is its special case,) or closure operator(in case of a Boolean algebra it is trivial one in which every subset is closed.)
To answer this question, we will. study the relationship between these structures. But, we will have to start from establishing some notational conventions
and from developing the conceptual framework of necessary definitions.
In addition to the notation commonly used in the literature of partiallyordered sets (or posets) and lattices, [5] the following conventions and simple
facts will be used hereafter.
If is a binary relation on the set X,
is its converse, and$R^{a}(A)=\{x\in X:\forall y\in A, yRx\},$ $R^{e}(A)=\{x\in X:\exists y\in A, yRx\}$ . We can simplify
our notation for single element subsets: $R(x)=R^{a}(\{x\})=R^{\epsilon}(\{x\})$ . In the caseof the partial order relation:$\leq^{a}(A)=\{x\in X:\forall y\in A, y\leq x\}$ ,$\leq^{e}(A)=\{x\in X:\exists y\in A, y\leq x\}$ ,$2^{3}$

$0$

$0$

$R$

$\leq*a(A)=\{x\in X:\forall y\in A, y\geq x\}$$\leq*e(A)=\{x\in X:\exists y\in A, y\geq x\}$$R^{*}$,.

65
Obviously, $R^{a}(A)=\cap\{R(x):x\in A\}$ and $R^{e}(A)=U\{R(x):x\in A\}$ .
If is a binary relation on set X, then the pair of functions from the power
set of X to itself $Aarrow R^{a}(A)$ and $Aarrow R^{*a}(A)$ forms a Galois connection, andtherefore both operations on subsets of X: $Aarrow R^{a}R^{*a}(A)$ and $Aarrow R^{*a}R^{a}(A)$are transitive closure operations on X understood as functions from the power:
set of X to itself such that for all $A,$
$R$

$f$

$B\subseteq X$

1. $A\subseteq f(A)$ ,2.

$A\subseteq B\Rightarrow f(A)\subseteq f(B)$, and

3. $f(A)=f(f(A))$ .
The third condition can be replaced by $A\subseteq f(B)\Rightarrow f(A)\subseteq f(B)$ ,Every closure operator is uniquely defined by the Moore family of its closed: A $=f(A)$ }, and every Moore family of subsets of X,
sets f-Cl{
i.e. family of sets which includes X and. is closed with respect to arbitrary
intersections, is the family of closed sets for the closure operator defined by $f(A)$$=\{B\in\Im;A\subseteq B\}$ . It is easy to see that for every closure operator its family ofclosed sets forms a complete lattice with respect to the set inclusion. Finally,
there is a natural partial ordering on closure operators defined by:$f\leq g$ if $\forall A\subseteq X:f(A)\subseteq g(A)$ ,which is equivalent to the condition for the families of closed subsets:
$=$

$\Im$

$A\subseteq X$$g- Cl\subseteq f- Cl$.

The history of the study of the relationship between partially ordered sets
and closure operators started from the work of Oystein Ore in which he observedthat there is a bijective correspondence between finite partially ordered sets andtopological spaces, i.e. finite spaces with closure operators satisfying
finitetwo additional conditions:$f(A)\cup f(B)=f(A\cup B)$ (the finite additivity of the closure
1. For all $A,$
operator distinguishing topological spaces).
2. For all $y\in.g,$ $x\in g(\{y\})\Rightarrow y\in f(\{x\})$ ( topology). [6]The correspondence is based on the relationship between the partial orderand the topological closure on singleton sets: $x\leq y$ iff $x\in f(\{y\})$ . This suffices
to define partial ordering when the closure operation is given. Here, the role ofcondition becomes clear, as otherwise the relation would be only reflexive
and transitive, i.e. a quasi-order (not necessarily anti-symmetric.)
Going the other direction, the extension of the closure operation from singletons to larger subsets can be achieved by:$f$$T_{0}$$B\subseteq X,$$T_{0}$$x,$$T_{0}$$f(A)=\leq^{*e}(A)=\cup\{\leq*(x):x\in A\}=\cup\{f(\{x\}):x\in A\}$.

The assumption that the posets and therefore top logical spaces under consideration are finite comes from the fact that topological spaces are defined by
the finite additivity condition (first condition above) which do not allow for theextension of the closure operation on singletons to infinite subsets. It is obvious,
topological spaces each singleton set is closed, yet thewhen we recall that inclosure is non trivial for infinite subsets.
$0$$T_{1}$

66
It is quite obvious that the topological closure operation above is only oneof many possible closure operations compatible with given partial order, i.e.
satisfying the condition: $x\leq y$ iff $x\in f(\{y\})$ . For our purpose the question what
are these compatible closure operations, and what conditions for the ordering
have to be added to identify the unique closure operation is of special interest.
Historically, the development of the inquiry was driven by different question.
The relationship between partially ordered sets and closure spaces provided a
way to embed the partially ordered set in a complete lattice of closed sets insuch a way that all existing finite or infinite infima and suprema of the poset
are preserved. These embeddings called completions by cuts are generalizationsof Dedekind’s construction of real numbers. Although, we are more interested
in the relationship between partial orders and closure operators rather, than in
the issue of embedding posets in complete lattices, we will study some moregeneral forms of completion, which will give as a point of departure for our own
inquiry. We have to recollect definitions of some of the concepts related to thissubject. [5]Definition 1

1. A nonvoid subset$\Rightarrow x\in J$.

$J$

of a poset $[P, \leq]$ is a semi-ideal if$\forall a\in J\forall x\in P,$$x\leq a$

2. A semi-ideal$J$

is principal if there exists $a\in P$ such that $J=\leq^{*}(a)$ .

3. A semi-ideal$J$

is an ideal if Va,

4. A semi-ideal$J$$\vee\{x:x\in A\}\in J$$b\in J,$ $a\vee b$

is a complete-ideal if

exists in$P\Rightarrow a\vee b\in J$.

$\vee\{x:x\in A\}exists$$\forall A\subseteq J,$.

in$P\Rightarrow$of a a poset $[P, \leq]$ is a closed-ideal if it contains all lower(and therefore
bounds to the set of its upper bounds, i.e.

5. A subset$J$$\leq^{*a}\leq^{a}(J)\subseteq J$$\leq^{*a}\leq a(J)=J.)$

It is obvious that every complete ideal is an ideal, every ideal is semi-ideal,
and that is a complete ideal. For finite posets there is no difference between
complete-ideals and ideals. Similarly, it is obvious that the families of completeideals, ideals, and semi-ideals are closed with respect to arbitrary intersections.
Thus they form Moore families, and they define transitive closure operators.
Semi-ideals are closed subsets for the closure operator:$f_{p}(A)=\leq^{*6}(A)=\{x\in X:\exists y\in A, y\geq x\}$ ,, complete
ideals are closed subsets for the closure operator and, and finally closed-ideals are closed sets for theideals for operator and$f_{c}(A)=\leq^{*a}\leq^{a}(A)$.
closure operator:
Proposition 1
Let $[P, \leq]$ be a poset and be a closure operator compatible with its partial$A=f(A)\Rightarrow A$ is$y\in P,$ $x\leq y$ordering, $i.e$ .
iff $x\in f(\{y\})$ . Thena semi-ideal.
$\leq^{*}(x)=f(\{x\})\subseteq f(A)=A$ , so
, and therefore
Proof:, i.e. A is a semi-ideal.
$P$$f_{i}$$f_{ci}$$f_{p}\leq f_{i}$$f_{1}\leq f_{ci}$$f$$\forall A\subseteq P,$$\forall x,$$\forall x\in A,$$\leq^{*e}(A)\subseteq A$$\leq^{*}(x)\subseteq A$

$,$

67
CorollaryFor every closure operator compatible with the partial order, $i.e$ . such that.
Vx, $y\in P,$
iff $x\in f(\{y\})$ , we haveThus, since $f_{c}(\{x\})=\leq^{*a}\leq^{a}(\{x\})=\leq^{*}(x)$ , every closed-ideal is a semi.
ideal, and therefore we have
Proposition 2
Let $[P, \leq]$ be a poset and be a closure operator compatible with its partial, and therefore weordering, $i.e$ . Vx, $y\in P,$ $x\leq y$ iff $x\in f(\{y\})$ . Then$f$$f_{p}\leq.f$$x\leq y$$f_{p}\leq f_{c}$$f$$f\leq f_{c}$.
haveProof: First observe that for every binary relation on set X, and for every)$A=R^{*a}(B)$ . ($A=R^{*a}R^{a}(A)$ iff
is simply $R^{a}(A)$ for$A=\leq^{*a}(B)=\{x\in
P:\forall
b\in
B,
x\in
f(\{b\})\}=$
Now, $f_{c}(A)=A$ iff$\cap\{f(\{b\}):b\in B\}$ , and therefore as an intersection of f-closed sets $f_{c}(A)$ must.
f-C1, which is equivalent tobe f-closed. Thus,$[P,\leq]$
is a complete lattice,
It can be easily shown that even when the poset
be replacedin general not all inequalities in the sequenceby equalities, although for obvious reason the middle inequality becomes anequality in finite posets. Simple example of the four element Boolean algebra(the diamond”) shows that to the set consisting of the two atoms (the middle
assigns
elements)
closure operator assigns as its closure all poset, while
the set of the three lower elements, so the first and the fourth closure operators
can be different. The example of the complete lattice of all natural numbers
ordered by divisibility, with the greatest element , the closure of the set of
all nonzero numbers is all poset for , but the set of all nonzero numbers isa semi-ideal and ideal, and therefore is closed for the first and second closure
operators. Only third inequality can be replaced by the equality in complete
lattices as the following proposition shows.
Proposition 3
If a poset $[P, \leq]$ is a $\omega mplete$ lattice, then $f_{ci}=f_{c}$ .
Proof: Let $J=f_{ci}(A)$ and $\vee\{x:x\in A\}=a$ , and $a\in J$ . But, by the definitionof the supremum of $A,$ $\leq^{a}(A)=\leq(c)$ , and therefore $\leq^{*a}\leq^{a}(A)=\leq^{*}(c)\subseteq J$ .
The reverse inclusion is always true, which gives us $f_{ci^{-}}C1=f_{c^{-}}C1$ .
We will return to the closure operators associated with posets which arecomplete lattices, but first we will focus on the poset completions, starting fromthe classical MacNeille theorem on the “completion by cuts. [5]Proposition 4 (MacNeille)a function ffom to the complete lattice
Let $[P, \leq]$ be a poset andof the -closed subsets of defined by $\varphi(x)=\leq^{*a}\leq^{a}(\{x\})$ . Then is aninjective, isotone and inverse-isotone function preserving all suprema and infima
that happen to enist in the poset $[P, \leq]$ .
Deflnition 2
Let $[P, \leq]$ be a poset and be an injective, isotone and inverse-isotonefunction from to a complete lattice satisfying the condition of minimality,
i.e. whose all complete-sublattices (i.e. substructures not only with respect tofinite, but also infinite infima and suprima) satisfy the condition:$f_{p}\leq f\leq f_{c}$$R$$A\subseteq X,$$B$$\exists B\subseteq X,$$\Rightarrow$$\exists B\subseteq P,$$f\leq f_{c}$$f_{c}-C1\subseteq$$f_{p}\leq f_{i}\leq f_{ci}\leq f_{c}can$$f_{p}$$f_{c}$$0$$f_{c}$$P$$\varphi$$P$$f_{\epsilon}$$\varphi$$\varphi$$P$$L_{c}$

$L$

$K$

68
$\varphi(P)\subseteq K\subseteq L\Rightarrow K=L$

. Such a complete lattice will be called a completionof the poset P. [7]Lemma 5
Let $[P, \leq]$ be a poset and be a transitive closure operator on subsets of P.
Then the function flom the poset to the complete latticeof the f-closed$\varphi(x)=f(\{x\})$subsets of given by
is injective, isotone and inverse-isotone iff
$f$

$P$

$\varphi$$L_{f}$$P$$\forall x\in P,$$f(\{x\})=f_{p}(\{x\})$.

$y\in P,$ $x\leq y$ iff
Proof: We want to show that [] iff$[\forall x\in P, f(\{x\})=f_{p}(\{x\})]$ . One direction of the implication is obvious. The
other can be shown when we recall that the three conditions for the transitive
closure operator are equivalent to:$\varphi(x)\subseteq$$\forall x,$$\forall A,$$B\subseteq P,$$A\subseteq f(B)$$\varphi(y)$iff $f(A)\subseteq f(B)$ .
iffiff$f(\{x\})\subseteq f(\{y\})$ iff $x\in f(\{y\})$ .
Thus,
Proposition 6
Let $[P, \leq]$ be a poset and be a transitive closure operator on subsets ofcompatible with the partial order, $i.e$ . Vx, $y\in P,$ $x\leq y$ iff $x\in f(\{y\})$ , which is$f(\{x\})=f_{p}(\{x\})$ . Then theequivalent to the condition
functi on fromto the complete latticeof the f-closed subsets of given by $\varphi(x)=f(\{x\})$defines a completion of the poset $[P, \leq]$ .
Proof: By Lemma 5, we have to show only thatis minimal. Suppose$K$
that there exists a complete-sublattice of, butsuch that. Then$f(A)=A$ , but A
different fromK. However, Va $\in A$ ,$f(\{a\})\in K$ and $f(\{a\})\subseteq A$ , and since A is f-closed, we have $\vee\{f(\{a\}):a\in A\}=$
A. But in $K$ andall suprema are identical, so $A\in K$ , contradiction.
Thus, we can see that all lattices of closed subsets with respect to closure
operators compatible with the partial order are completions of poset $[P, \leq]$ .
However, MacNeille’s completion has the advantage that it preserves all existing
infima and suprema of the poset. Also, it has been known for long time that
MacNeille’s completion of Boolean algebra is a Boolean algebra, while it does
not have to be true for ideal completion defined by . On the other hand,
the ideal completions preserve distributivity and modularity of lattices, whIle
MacNeille’s completion not necessarily. [5]The strongest argument for the superiority of MacNeille’s completion comesfrom the fact that it is isomorphic to the original poset whenever it is alreadya complete lattice, while for example the semi-ideal completion defined by isnever isomorphic, unless the poset is a chain. The first fact follows easily fromthe following proposition. [5]Proposition 7
A poset $[P, \leq]$ is a complete lattice iff all its closed-ideals, $i.e$ . subsets withrespect to the dosure operator are rincipal ideals.
The second fact, that no complete lattice is isomorphic to the lattice of
its semi-ideals (subsets closed with respect to ), unless it is a chain follows
from the simple fact that the union of any two principal semi-ideals
andsemi-ideal,beisomorphic
and for the lattice tois awith its semi-ideal
completion this semi-deal has to be principal semi-ideal. But this means$\forall x,y\in P,$$x\in f_{p}(\{y\})$$x\leq y$

$f$

$P$

$\forall x\in P,$$P$$\varphi$$P$$L_{f}$$L_{f}$$\varphi(P)\subseteq K\subseteq L_{f}$$L_{f)}$$L_{f}$$\exists A\subseteq P,$$\not\in$$L_{f}$$f_{i}$$f_{p}$$f_{c}$$p$$f_{p}$$\leq^{*}(x)$$\leq^{*}(y)$$\leq^{*}(z)$69that has to belong to one or the other of. Therefore, has
andto be either or , so either.
orThus, the only complete lattices isomorphic to their semi-ideal completions
are complete chains, and for complete chains.
Thus, we have reasons to believe that the closure operator
is a superior
candidate among the closure operators compatible with the given partial order considered above for our purpose of modeling integration of information.
However, there are many other possible closure operators compatible with thepartial order which we did not consider yet. There is no convincing argument
for the choice of closed-ideal closure operator.
3. Transitive closure operators compatible with structures built
over the partIal orderFirst, let’s recall that the closure operator has been constructed with thehelp of the Galois connection defined by the functions on the power set of theset :$Aarrow R^{a}(A)$ and $Aarrow R^{*a}(A)$ . In this particular case the relation
was thepartial order of the poset. We can as how to identify those closure operators
which are defined by some binary relation R. An answer considering symmetric
relations was given by Oystein Ore in his early study of Galois connections. [8]Definition 3
A function from a poset $[P, \leq]$ to poset $[Q, \leq]$ is called a dual $isomo\varphi hism$,
if it is bijective, and satisfies the following two conditions:i) Vx, $y\in P,$
,
$y\in
P,$
ii) Vx,.
A dual isomorphism from a poset $[P, \leq]$ to itself is called an involution, if it$\gamma\gamma(x)=x$ .
is of order two, i.e. if
An orthocomplementation on a poset $[P, \leq]$ with the least element and thegreatest element 1 is an involutive dual isomorphism, i.e. a bijective mappingfrom a poset$[P, \leq]$ to itself
such that Vx, $y\in P$ :i) $\gamma\gamma(x)=x$ ,ii) if $x\leq y$, then,$x\wedge\gamma(x)=0$
iii)
and $x\vee\gamma(a)=1$ , whenever the meet and join exist.
In usual notation
is indicated by $x’$ .
Frequently, the fact that $a\leq b$ ’ is written
and is read “a is orthogonal
to . A poset with orthocomplementation is called an orthoposet, a latticewith an orthocomplementation is called an ortholattice.
The following theorem belonging to the earliest studies of Galois connections
provides the condition for the closure operation to be a Galois closure operation,i.e. closure defined by a Galois connection. [8]Proposition 8
Let be a transitive closure operator on set P. Then there is a binary, symmetric relation
on , such that is a Galois closure operator defined by$Aarrow R^{a}R^{a}(A)$iff there vists an involution on the complete lattice of all f$R(x)=$dosed $sub_{8}ets$ . For the given closure operator and involution$\leq^{*}(x)$

$z$

$x$

$x\leq y$$y$$\leq^{*}(y)$$z$$y\leq x$$f_{p}=f_{c}$$f_{c}$$f_{c}$

$P$

$R$

$k$

$\gamma$$x\leq y\Rightarrow\gamma(y)\leq\gamma(x)$$\gamma(x)\leq\gamma(y)\Rightarrow y\leq x$$\forall x\in P,$$0$$\gamma$$x\prec\gamma(x)$$\gamma(y)\leq\gamma(x)$$\gamma(x)$$a\perp b$

$b$

$f$

$R$

$P$

$e$

$f$

$\gamma$$f$$\gamma(f(\{x\})$.

$\gamma,$$\forall x\in P,$

70
is anti-reflexive ( $xRx$ for no in P) or satisfies the weaker$xRx\Rightarrow xRy$
condition:
for all in , then the involution defines anorthocomplementation on the lattice of f-closed subsets.
It turns out that the Galois closures for the partial order and orthogonality
relation of the orthoposet coincide.
Proposition 9$\leq^{*a}\leq^{a}(A)=$Let $[P, \leq, xarrow\gamma(x)=x’]$ be an orthoposet. Then

If the relation

$R$

$x$

$\forall x\in P,$

$P$

$y$

$\forall A\subseteq P,$$\cdot$$\perp^{a}\perp a(A)$.

Proof: $x\in 1^{a}1^{a}(A)$ iff $\{\forall y\in P, [\forall a\in A, y\perp a]\Rightarrow x\perp y\}$ iff$\{\forall y\in P, [\forall a\in A, y\leq a’]\Rightarrow x\leq y’\}$ iff$\{\forall y\in P, [\forall a\in A, a\leq y’]\Rightarrow x\leq y’\}$ iff $x\in\leq*a\leq^{a}(A)$ .
Thus, when we have an orthoposet, instead of just a poset, the closure operator defined by closed-ideals associated with this structure becomes uniquelydetermined. What is interesting, that this way we are also closer to the quantum logic formalism of quantum mechanics based on the concept of an ortholattice satisfying additional conditions of being orthomodular [if $a\leq b$ , then$=a\vee(b\wedge a’)]$ , complete, atomic and atomistic lattice with the atomic covering
property and exchange property (orthomodular, complete, atomic, AC lattice if
the redundant conditions have been eliminated). [1]We can consider more general structure of a poset (not orthoposet) with socalled strong orthogonality $relation\perp defined$ on by the conditions:
$b$

$P$

1. The $relation\perp is$ symmetric,
2.

$\forall x\in P,$

3.

$\forall x,$$x\perp x\Rightarrow x\perp y$for all

$y$

in ,
$P$

$y\in P,$ $x\leq yiff\perp(y)\subseteq\perp(x)$.

The same $symbol\perp is$ used here, as every orthogonality relation defined by or) is a special instance of strong orthogonality.
if
thocomplementation (Although the Galois closure defined by the strong orthogonality relation does
not coincide in general with the closed-ideal closure operator, it is compatible
with the partial order.
Proposition 10
be a poset with strong orthogonality relation. Then the closureLet
operator defined on subsets A of by $f(A)=\perp^{a}1^{a}(A)$ is compatible with the$f(\{x\})=f_{p}(\{x\})$ .
partial order, $i.e$ .
iff
[
iff
Proof:
iff $y\in f(\{x\})$ ].
.
Now we need to show only
For every relation and every subset , we have $R^{a}R^{*a}R^{a}(A)=R^{a}(A)$ ,but the orthogonality is symmetric, so$a\perp b$$a\leq b’$$[P, \leq, \perp]$

$P$

$f$

$\forall x\in P,$$\forall x,y\in P,$$y\leq x$$y\in f_{p}(\{x\})$$\perp(x)\subseteq\perp(y)\Rightarrow\perp a\perp a(x)\subseteq\perp a\perp a(y)$$\forall x\in P,$

$A$

$R$

$1^{a}\perp a\perp a(A)=\perp a(A)$Thus,$\forall x,$$y\in P,$$f(\{x\})\subseteq f_{p}(\{x\})$.

[ $y\in f(\{x\})$ iff$y\in 1^{a}\perp(x)\Rightarrow$$\perp(x)\subseteq\perp(y)\Rightarrow$.
iff
In either case we can use the orthogonality relation to determine selectionof the closure operation out of those compatible with the partial order.
$y\leq x$$y\in f_{p}(\{x\})$71
4. Conclusion
For given partially ordered sets, there are many different closure operators
compatible with the order. There are some reasons why some of these closures
may be more useful, but there is no objective criterion for the selection, evenwhen we think in terms of the modeling information integration. As it has beenshown, the structure of an orthoposet, or more generally of a poset with strong
orthogonality relation on a set , which can be interpreted as a generalized“logic” for information integration, can be used as a means to select a uniqueclosure operation on P. As a result, we get a bijective correspondence between
posets with strong orthogonality relation and closure operators.
This concludes the first stage of our inquiry. The abstract orthogonality
relations defined on by the conditions:
1. The $relation\perp is$ symmetric,
for all in ,2.
correspond to weak tolerance relations, which are simply complement rela). $[9]$ Since tolerance relations
tions to orthogonality relations ( $xTy$ iff not
are mathematical formalizations of similarity, or the relation which is frequently
invoked as Wittgenstein’s (family resemblance,” there is an interesting question
about the function of this relation in the model of information integration. From
the fact that equivalence relations are just transitive tolerances, we can expect
some relevance of these relations for the process of abstraction of information.
The next step is to $co$ nsider closure operations related to partially orderedsets, but defined not on the entire set on which order is defined, but on theirsubsets, such as sets of atoms, join-irreducible elements, etc.
The consecutive step is to search for the conditions for irreducibility of thestructures describing information integration which is our ultimate goal. Thiscan be done either in terms of partial order, or of closure operations.
$P$

$P$

$\forall x\in P,$$x\perp x\Rightarrow x\perp y$

$y$

$P$

$x\perp y$

References[1] M. J. Schroeder, Model of structural information based on the latticeof closed subsets. In Y. Kobayashi, T. Adachi (eds) Proceedings of The TenthSymposium on Algebra, Languages, and Computation, Toho University, 2006,

pp.32-47.
[2] M. J. Schroeder, Philosophical Foundations for the Concept of Information: Selective and Structural Information. In Proceedings of the Third International Conference on the Foundations of Information Science, Paris 2005.

http: $//w^{r}ww.mdpi.org/fls2005$ .
[3] R. Penrose, Shadows of the Mind: A Search for the Missing Science ofConsciousness. Oxford University Press, Oxford, 1994.
[4] J. M. Jauch, Foundations of Quantum Mechanics. Addison-Wesley, Reading, Mass. 1968.
. ed. American Mathematical Society[5] G. Birkhoff, Lattice Theory,
Colloquium Publications, Vol XXV, Providence, R. I., 1967.
[6] O. Ore, Some Studies On Closure Relations. Duke Math. J. , 10 (1943),$3^{rd}$761-785.

72[7] W. R. Tunnicliffe, The completion of a partially ordered set with respect
to a polarization. Proc. London Math. Soc., (3) 28 (1974), 13-27.
[8] O. Ore, Galois connexions. $\pi ans$ . Amer. Math. Soc., 55 (1944), 493-

513.

[9] M. J. Schroeder, M. H. Wright, Tolerance and weak tolerance relations,J. Combin. Math. and Combin. Comput., 11(1992), 123-160.