Logical and algebraic properties of generalized orthomodular posets
aa r X i v : . [ m a t h . L O ] S e p Logical and algebraic properties of generalizedorthomodular posets
Ivan Chajda and Helmut L¨anger
Abstract
Generalized orthomodular posets were introduced recently by D. Fazio, A. Leddaand the first author of the present paper in order to establish a useful tool forstudying the logic of quantum mechanics. They investigated structural propertiesof these posets. In the present paper we study logical and algebraic properties ofthese posets. In particular, we investigate conditions under which they can be con-verted into operator residuated structures. Further, we study their representationby means of algebras (directoids) with everywhere defined operations. We provecongruence properties for the class of algebras assigned to generalized orthomodu-lar posets and, in particular, for a subvariety of this class determined by a simpleidentity. Finally, in contrast to the fact that the Dedekind-MacNeille completionof an orthomodular poset need not be an orthomodular lattice we show that theDedekind-MacNeille completion of a stronger version of a generalized orthomodularposet is nearly an orthomodular lattice.
AMS Subject Classification:
Keywords: generalized orthomodular poset, orthomodular poset, orthomodular lattice,strong generalized orthomodular poset, assigned directoid, conditional operator residua-tion, operator residuation, congruence distributivity, congruence permutability, congru-ence regularity, Dedekind-MacNeille completion
Although the logic of quantum mechanics was axiomatized by K. Husimi ([13]) andG. Birkhoff and J. von Neumann ([1]) by means of orthomodular lattices, it was earlyshown that this description need not be appropriate in all concerns. Orthomodularlattices characterize the lattice of projection operators on a Hilbert space. In 1963,many years after orthomodular lattices have been introduced, it was realized that amore appropriate formalization of the logic of quantum mechanics could be obtained byreplacing the axioms of orthomodular lattices by the weaker axioms of orthomodularposets, see e.g. [15]. The reason for this weakening was that in the logic of quantum Support of the research of the authors by the Austrian Science Fund (FWF), project I 4579-N, and theCzech Science Foundation (GA ˇCR), project 20-09869L, entitled “The many facets of orthomodularity”,as well as by ¨OAD, project CZ 02/2019, entitled “Function algebras and ordered structures related tologic and data fusion”, and, concerning the first author, by IGA, project PˇrF 2020 014, is gratefullyacknowledged. generalized orthomodular posets was introduced in [4]. In that paper theorder-theoretical properties of generalized orthomodular posets were investigated.Since all so-called quantum structures, such as orthomodular lattices, orthomodularposets, generalized orthomodular posets (see also [9] and [10]), are assumed to be analgebraic axiomatization of the semantics of the logic of quantum mechanics, the naturalquestion arises how the logical connective of implication should be modeled within theselogics. Implication turns out to be one of the most fundamental and most productive log-ical connectives which enables logical deduction and therefore it should be introduced ina way acceptable in logics. Usually, implication is considered to be sound if it is relatedwith conjunction via the so-called adjointness , i.e. implication and conjunction shouldform a residuated pair . For orthomodular lattices this task was solved by the authors in[7] and [8], for orthomodular posets in [6] and for some more general posets in [10].The aim of the present paper is to describe some algebraic and logical properties ofgeneralized orthomodular posets, to show how these posets can be represented by meansof algebras with everywhere defined operations and to introduce certain modificationsof the connective of implication related to certain types of conjunction via a generalizedversion of adjointness.A certain generalization of orthomodular posets avoiding existential quantifiers was intro-duced in [4] under the name pseudo-orthomodular poset where it was shown that thoughsuch a poset cannot be organized into a residuated structure, it is possible to define bi-nary operators M ( x, y ) and R ( x, y ) on such a poset satisfying the so-called operator leftadjointness(1) M ( x, y ) ⊆ L ( z ) if and only if L ( x ) ⊆ R ( y, z ).This motivated us to find something analogous for generalized orthomodular posets. Sincethe definition of a generalized orthomodular poset is a bit more simple than that of apseudo-orthomodular one, we will need only one operator, namely R ( x, y ). On the otherhand, analogously as in [6], we need an additional condition guaranteeing property (1).Since our operator M ( x, y ) will be commutative, instead of operator left adjointnesswe will have conditional operator adjointness. Further, we will show that if a stronggeneralized orthomodular poset is considered then the conjunction can be defined in aslightly different way as well as the residuated operator R ( x, y ) such that we really obtaina left residuated structure. We start by defining the aforementioned concepts.Let ( P, ≤ ) be a poset, a, b ∈ P and A, B ⊆ P . Then A ≤ B should mean x ≤ y for all x ∈ A and y ∈ B . Instead of { a } ≤ B , A ≤ { b } and { a } ≤ { b } we simply write a ≤ B ,2 ≤ b and a ≤ b , respectively. Moreover, we define L ( A ) := { x ∈ P | x ≤ A } ,U ( A ) := { x ∈ P | A ≤ x } and call these sets the lower and upper cone of A , respectively. Instead of L ( { a, b } ), L ( { a } ∪ B ), L ( A ∪ B ) and L ( U ( A )) we simply write L ( a, b ), L ( a, B ), L ( A, B ) and LU ( A ),respectively. Analogously we proceed in similar cases.Let ( P, ≤ , ′ ) be a poset with a unary operation ′ and A ⊆ P . We define A ′ := { x ′ | x ∈ A } .We call ′ an antitone involution of ( P, ≤ ) if both x ′′ ≈ x and if for x, y ∈ P , x ≤ y implies y ′ ≤ x ′ . Let ( P, ≤ , ′ , ,
1) be a bounded poset with a unary operation ′ . We call ′ a complementation on ( P, ≤ , ,
1) if L ( x, x ′ ) ≈ { } and U ( x, x ′ ) ≈ { } .An orthoposet is a bounded poset ( P, ≤ , ′ , ,
1) with an antitone involution ′ which is acomplementation.Any orthoposet ( P, ≤ , ′ , ,
1) satisfies the
De Morgan’s laws ( L ( x, y )) ′ ≈ U ( x ′ , y ′ ) , ( U ( x, y )) ′ ≈ L ( x ′ , y ′ ) . The following concept was introduced in [4].
Definition 2.1. A generalized orthomodular poset is an orthoposet ( P, ≤ , ′ , , satisfy-ing the condition (2) x ≤ y implies U ( y ) = U ( x, L ( x ′ , y )) . Using De Morgan’s laws it is elementary to prove that (2) is equivalent to(3) x ≤ y implies L ( x ) = L ( y, U ( x, y ′ )).Since y is the smallest element of U ( y ), U ( y ) = U ( x, L ( x ′ , y )) means that y is the smallestelement of U ( x, L ( x ′ , y )), i.e. y is the smallest upper bound, i.e. the supremum, of { x } ∪ L ( x ′ , y ) which means y = x ∨ L ( x ′ , y ). Hence (2) can be written in the form x ≤ y implies y = x ∨ L ( x ′ , y )) . Analogously, (3) can be written in the form x ≤ y implies x = y ∧ U ( x, y ′ ) . Now we define also a stronger version of a generalized orthomodular poset as follows.
Definition 2.2. A strong generalized orthomodular poset is an orthoposet ( P, ≤ , ′ , , satisfying the condition that for all x ∈ P and for all subsets B of P (4) x ≤ U ( B ) implies U ( B ) = U ( x, L ( x ′ , U ( B ))) . Using De Morgan’s laws it is elementary to prove that this is equivalent to the conditionthat for all y ∈ P and A ⊆ P L ( A ) ≤ y implies L ( A ) = L ( y, U ( L ( A ) , y ′ )).Obviously, every strong generalized orthomodular poset is a generalized orthomodularposet.Now we recall the useful concept of a directoid which serves as an algebraization of agiven poset. In Section 4 we will show how a generalized orthomodular poset can beconverted into an algebra with everywhere defined operations by using of an assigneddirectoid.A ( join- ) directoid (see [5] and [14]) is a groupoid ( D, ⊔ ) satisfying the following identities:(i) x ⊔ x ≈ x (idempotency),(ii) x ⊔ y ≈ y ⊔ x (commutativity),(iii) x ⊔ (( x ⊔ y ) ⊔ z ) ≈ ( x ⊔ y ) ⊔ z (weak associativity).Let P = ( P, ≤ ) be a poset. A groupoid ( P, ⊔ ) is called a directoid assigned to P if itsatisfies the following conditions for all x, y ∈ P : • x ⊔ y = y if x ≤ y , • x ⊔ y = y ⊔ x ∈ U ( x, y ).Assume D ( P ) to be a directoid assigned to P . Then D ( P ) is a directoid by the abovedefinition. Conversely, let D = ( D, ⊔ ) be a directoid and let P ( D ) the ordered pair ( D, ≤ )where ≤ denotes the binary relation on D defined by x ≤ y if x ⊔ y = y, the so-called induced order . Then P ( D ) is a poset and P ( D ( P )) = P . This shows that,though D ( P ) is in general not uniquely determined by P , it contains the whole informationon P . If P = ( P, ≤ , ,
1) is a bounded poset then there exists an assigned directoid since1 ∈ U ( x, y ) and hence U ( x, y ) = ∅ for all x, y ∈ P . One can easily check that an assigneddirectoid D ( P ) = ( P, ⊔ , ,
1) satisfies the identities x ⊔ ≈ x and x ⊔ ≈ . Such a directoid will be referred to as a bounded directoid . Now we introduce one of our main concepts.
Definition 3.1. A conditionally operator residuated poset is an ordered six-tuple R =( P, ≤ , ′ , R, , such that ( P, ≤ , ′ , , is a bounded poset with a unary antitone operation ′ and R is a mapping from P to P satisfying the following conditions for all x, y, z ∈ P : (i) If x ′ ≤ y then L ( x, y ) ⊆ L ( z ) implies L ( x ) ⊆ R ( y, z ) , if z ≤ y then L ( x ) ⊆ R ( y, z ) implies L ( x, y ) ⊆ L ( z ) , (iii) R ( x, ≈ L ( x ′ ) , (iv) R ( x ′′ , x ) ≈ P . R is said to satisfy operator divisibility if x ≤ y implies L ( y, U ( R ( y, x ))) = L ( x ) . In the sequel we will show that there are close connections between generalized ortho-modular posets and conditionally operator residuated posets.
Theorem 3.2.
Let ( P, ≤ , ′ , , be a generalized orthomodular poset and put R ( x, y ) := LU ( x ′ , y ) for all x, y ∈ P. Then R := ( P, ≤ , ′ , R, , is a conditionally operator residuated poset satisfying operatordivisibility.Proof. Let a, b, c ∈ P .(i) If a ′ ≤ b and L ( a, b ) ⊆ L ( c ) then b ′ ≤ a and using (2), we compute L ( a ) = LU ( a ) = LU ( b ′ , L ( a, b )) ⊆ LU ( b ′ , L ( c )) = LU ( b ′ , c ) = R ( b, c ) . (ii) If c ≤ b and L ( a ) ⊆ R ( b, c ) then, using (3), we derive L ( a, b ) = L ( a ) ∩ L ( b ) ⊆ LU ( b ′ , c ) ∩ L ( b ) = L ( U ( b ′ , c ) , b ) = L ( c ) . (iii) Further, we have R ( x, ≈ LU ( x ′ , ≈ LU ( x ′ ) ≈ L ( x ′ ).(iv) We have R ( x ′′ , x ) ≈ LU ( x ′′′ , x ) ≈ LU ( x ′ , x ) ≈ L (1) = P .In case a ≤ b we finally have L ( b, U ( R ( b, a ))) = L ( b, U LU ( b ′ , a )) = L ( b, U ( b ′ , a )) = L ( a ) . Thus R satisfies operator divisibility.Hence we have shown that every generalized orthomodular poset can be organized intoa conditionally operator residuated poset in analogy to the fact that every orthomod-ular poset can be converted into a conditionally residuated poset (cf. [6]). Let us notethat a generalized orthomodular poset can be reduced to an orthomodular poset if it isorthogonal (see [2] for this concept and its properties).Now we are interested in the converse question, i.e. whether a conditionally operatorresiduated poset is in fact a generalized orthomodular poset. In the next theorem we showthat this is the case if the unary operation is antitone and satisfies operator divisibility.5 heorem 3.3. Let ( P, ≤ , ′ , R, , be a conditionally operator residuated poset satisfyingoperator divisibility and assume R ( x, y ) = LU ( x ′ , y ) for all x, y ∈ P. Then P := ( P, ≤ , ′ , , is a generalized orthomodular poset.Proof. Let a, b ∈ P . From L ( a ′ ) ⊆ R ( a,
0) we obtain L ( a ′ , a ) ⊆ L (0), i.e. L ( a, a ′ ) = { } .From L ( a, a ′ ) ⊆ L (0) we obtain a ∈ L ( a ) ⊆ R ( a ′ ,
0) = L ( a ′′ ) by (iii) which yields a ≤ a ′′ .Conversely, we apply (iv) and from L ( a ′′ ) ⊆ R ( a ′′ , a ) we obtain a ′′ ∈ L ( a ′′ , a ′′ ) ⊆ L ( a )which yields a ′′ ≤ a . Together we have a ′′ = a . Thus ′ is an involution. Hence we canapply De Morgan’s laws to L ( a, a ′ ) = { } in order to obtain U ( a, a ′ ) = { } , i.e. ′ is also acomplementation on ( P, ≤ ) and hence P an orthoposet. If a ≤ b then using De Morgan’slaws and operator divisibility we finally obtain U ( a, L ( a ′ , b )) = ( L ( a ′ , U ( a, b ′ ))) ′ = ( L ( a ′ , U LU ( a, b ′ ))) ′ = ( L ( a ′ , U ( R ( a ′ , b ′ )))) ′ == ( L ( b ′ )) ′ = U ( b ) . Altogether, P is a generalized orthomodular poset.It is a natural question whether we can obtain also a structure which is residuated in abroader sense but no additional conditions must be supposed.In what follows, we show that strong generalized orthomodular posets can be convertedinto left residuated structures where both the operators M ( x, y ) and R ( x, y ) (i.e. con-junction and implication in a broad sense) are everywhere defined. For this, we modifyDefinition 3.1 as follows. Definition 3.4. An operator residuated poset is an ordered seven-tuple R = ( P, ≤ , ′ , M, R, , such that ( P, ≤ , ′ , , is a bounded poset with a unary antitone operation ′ and M and R are mappings from P to P satisfying the following conditions for all x, y, z ∈ P : (i) M ( x, y ) ⊆ L ( z ) if and only if L ( x ) ⊆ R ( y, z ) (operator adjointness) , (ii) R ( x, ≈ L ( x ′ ) , (iii) R ( x, x ′′ ) ≈ R ( x ′′ , x ) ≈ P . R is said to satisfy operator divisibility if x ≤ y implies L ( y, U ( R ( y, x ))) = L ( x ) . Now we can show that strong generalized orthomodular posets can be organized intoleft residuated structures analogously as it was done for modular lattices and stronglymodular posets in [11] and [12].
Theorem 3.5.
Let ( P, ≤ , ′ , , be a strong generalized orthomodular poset and put M ( x, y ) := L ( U ( x, y ′ ) , y ) ,R ( x, y ) := LU ( x ′ , L ( x, y )) for all x, y ∈ P . Then R := ( P, ≤ , ′ , M, R, , is an operator residuated poset satisfyingoperator divisibility. roof. Let a, b, c ∈ P .(i) If M ( a, b ) ⊆ L ( c ) then, using (4), we compute L ( a ) = LU ( a ) ⊆ LU ( a, b ′ ) = LU ( b ′ , L ( U ( a, b ′ ) , b )) = LU ( b ′ , L ( b ) ∩ L ( U ( a, b ′ ) , b )) ⊆⊆ LU ( b ′ , L ( b ) ∩ L ( c )) = LU ( b ′ , L ( b, c )) = R ( b, c ) . If, conversely, L ( a ) ⊆ R ( b, c ) then U ( b ′ , L ( b, c )) = U LU ( b ′ , L ( b, c )) = U ( R ( b, c )) ⊆ U L ( a ) = U ( a )and hence, using (5), we obtain M ( a, b ) = L ( U ( a, b ′ ) , b ) = L ( U ( a ) ∩ U ( b ′ ) , b ) ⊆ L ( U ( b ′ , L ( b, c )) ∩ U ( b ′ ) , b ) == L ( U ( b ′ , L ( b, c )) , b ) = L ( b, c ) ⊆ L ( c ) . It is easy to verify the remaining conditions from Definition 3.4.(ii) We have R ( x, ≈ LU ( x ′ , L ( x, ≈ LU ( x ′ ) ≈ L ( x ′ ).(iii) We have R ( x, x ′′ ) ≈ R ( x ′′ , x ) ≈ R ( x, x ) ≈ LU ( x ′ , L ( x, x )) ≈ LU ( x ′ , x ) ≈ L (1) ≈ P .Finally, in case a ≤ b we have L ( b, U ( R ( b, a ))) = L ( b, U LU ( b ′ , L ( b, a ))) = L ( b, U ( b ′ , a )) = L ( a ) , thus R satisfies operator divisibility.Also some kind of a converse of the previous result holds. Theorem 3.6.
Let ( P, ≤ , ′ , M, R, , be an operator residuated poset satisfying operatordivisibility and assume M ( x, y ) = L ( U ( x, y ′ ) , y ) ,R ( x, y ) = LU ( x ′ , L ( x, y )) for all x, y ∈ P . Then P := ( P, ≤ , ′ , , is a generalized orthomodular poset.Proof. Let a, b ∈ P . We have M ( x, x ) ≈ L ( U ( x, x ′ ) , x ) ≈ L ( x ) ,M ( x ′ , x ) ≈ L ( U ( x ′ , x ′ ) , x ) ≈ L ( x, x ′ ) . From L ( a ′ ) ⊆ R ( a,
0) we obtain L ( a, a ′ ) = M ( a ′ , a ) ⊆ L (0), i.e. L ( a, a ′ ) = { } . From L ( a ) ⊆ R ( a, a ′′ ) we obtain a ∈ L ( a ) = M ( a, a ) ⊆ L ( a ′′ ) which implies a ≤ a ′′ . Conversely,From L ( a ′′ ) ⊆ R ( a ′′ , a ) we obtain a ′′ ∈ L ( a ′′ ) = M ( a ′′ , a ′′ ) ⊆ L ( a ) which implies a ′′ ≤ a .Together we have a ′′ = a . Thus ′ is an involution. Hence we can apply De Morgan’s lawsto L ( a, a ′ ) = { } in order to obtain U ( a, a ′ ) = { } , i.e. ′ is also a complementation on( P, ≤ ) and hence P an orthoposet. If a ≤ b then using De Morgan’s laws and operatordivisibility we finally obtain U ( a, L ( a ′ , b )) = ( L ( a ′ , U ( a, b ′ ))) ′ = ( L ( a ′ , U LU ( a, L ( a ′ , b ′ )))) ′ = ( L ( a ′ , U ( R ( a ′ , b ′ )))) ′ == ( L ( b ′ )) ′ = U ( b ) , i.e. P is a generalized orthomodular poset. 7 Directoids assigned to generalized orthomodularposets
Let P = ( P, ≤ , ′ , ,
1) be a bounded poset with a unary operation ′ . An algebra ( P, ⊔ , ′ , ,
1) of type (2 , , ,
0) is called assigned to P if it satisfies the following conditions for all x, y ∈ P : • x ⊔ y = y if x ≤ y , • x ⊔ y = y ⊔ x ∈ U ( x, y ).Let A = ( A, ⊔ , ′ , ,
1) be an algebra of type (2 , , , x ′ ⊔ y ′ ) ′ by x ⊓ y andlet G ( A ) denote the ordered quintuple ( A, ≤ , ′ , ,
1) where ≤ denotes the binary relationon A defined by x ≤ y if x ⊔ y = y. If P = ( P, ≤ , ′ , ,
1) is a generalized orthomodular poset and A ( P ) = ( P, ⊔ , ′ , ,
1) analgebra assigned to P then ( P, ⊔ ) is a directoid assigned to the poset ( P, ≤ ) and hence A ( P ) will be called a directoid assigned to P .As promised in Section 2, we can characterize the class of all generalized orthomodularposets by means of assigned directoids. At first, we state the following Lemma. Lemma 4.1.
Let P = ( P, ≤ , ′ , , be a generalized orthomodular poset, a, b ∈ P and A ( P ) = ( P, ⊔ , ′ , , a directoid assigned to P . Then (i) ( P, ⊔ ) is a directoid, (ii) G ( A ( P )) = P , (iii) L ( a, b ) = { ( a ⊓ x ) ⊓ ( b ⊓ x ) | x ∈ P } = { x ∈ P | ( a ⊓ x ) ⊓ ( b ⊓ x ) = x } , (iv) U ( a, b ) = { ( a ⊔ x ) ⊔ ( b ⊔ x ) | x ∈ P } = { x ∈ P | ( a ⊔ x ) ⊔ ( b ⊔ x ) = x } .Proof. (i) and (ii) are already shown in Section 2. We prove the remaining assertions.(iii) If c ∈ P then ( a ⊓ c ) ⊓ ( b ⊓ c ) ≤ a ⊓ c ≤ a, ( a ⊓ c ) ⊓ ( b ⊓ c ) ≤ b ⊓ c ≤ b, i.e. ( a ⊓ c ) ⊓ ( b ⊓ c ) ∈ L ( a, b ). If, conversely, c ∈ L ( a, b ) then( a ⊓ c ) ⊓ ( b ⊓ c ) = c ⊓ c = c. (iv) This follows from (iii) by duality. 8ondition (ii) of Lemma 4.1 shows that, though A ( P ) is in general not uniquely deter-mined by P , it contains the whole information on P , i.e. P can be reconstructed from A ( P ).Let P = ( P, ≤ , ′ , ,
1) be a generalized orthomodular poset and A ( P ) = ( P, ⊔ , ′ , ,
1) adirectoid assigned to P . Then we can easily check that the following conditions hold forall x, y ∈ P : • x ⊓ y = x if x ≤ y , • x ⊓ y = y ⊓ x ∈ L ( x, y ), • ( x ⊔ y ) ⊓ x ≈ x , • ( x ⊓ y ) ⊔ x ≈ x .Moreover, ⊓ satisfies identities (i) – (iii) from Section 2 (with ⊔ replaced by ⊓ ) and hence( P, ⊓ ) is called a ( meet- ) directoid .We now want to describe those directoids which are assigned to generalized orthomodularposets. We present here an easy characterization using three identities and one implica-tion. The crucial thing is that we characterize posets by algebras with everywhere definedoperations. Theorem 4.2.
Let P = ( P, ≤ , ′ , , be a bounded poset with a unary operation ′ and A ( P ) = ( P, ⊔ , ′ , , an algebra assigned to P . Then P is a generalized orthomodularposet if and only if A ( P ) satisfies the following conditions: (i) if ( x ⊔ z ) ⊔ ((( x ′ ⊓ w ) ⊓ (( x ⊔ y ) ⊓ w )) ⊔ z ) = z for all w ∈ P then ( x ⊔ y ) ⊔ z = z , (ii) ( x ⊓ y ) ⊔ x ≈ x , (iii) ( x ⊔ y ) ⊔ ( x ′ ⊔ y ) ≈ , (iv) x ′′ ≈ x .Proof. First assume P to be a generalized orthomodular poset. According to Lemma 4.1, L ( x ′ , x ⊔ y ) = { ( x ′ ⊓ w ) ⊓ (( x ⊔ y ) ⊓ w ) | w ∈ P } . Moreover, U ( x, L ( x ′ , x ⊔ y )) = { z ∈ P | ( x ⊔ z ) ⊔ ( w ⊔ z ) = z for all w ∈ L ( x ′ , x ⊔ y ) } . Now (i) follows from U ( x, L ( x ′ , x ⊔ y )) ⊆ U ( x ⊔ y ). Identity (ii) follows from x ⊓ y ≤ x andidentity (iii) from ( x ⊔ y ) ⊔ ( x ′ ⊔ y ) ∈ U ( x, x ′ ) = { } . Identity (iv) is evident. Conversely,assume A ( P ) to satisfy (i) – (iv). If we substitute x and y in (ii) by x ′ and y ′ , respectively,and apply (iv) then we obtain that x ≤ y implies y ′ ≤ y ′ ⊔ x ′ = ( x ⊔ y ) ′ ⊔ x ′ = ( x ′′ ⊔ y ′′ ) ′ ⊔ x ′ = ( x ′ ⊓ y ′ ) ⊔ x ′ = x ′ , i.e. ′ is antitone. Because of (iv), ′ is an involution. Altogether, ′ is an antitone involutionon ( P, ≤ ). If y ∈ U ( x, x ′ ) then x, x ′ ≤ z , thus x ⊔ z = z = x ′ ⊔ z . Using (v) we compute y = y ⊔ y = ( x ⊔ y ) ⊔ ( x ′ ⊔ y ) = 19roving U ( x, x ′ ) = { } for all x ∈ P . Due to De Morgan’s laws we have L ( x, x ′ ) = { } showing that x ′ is a complement of x . Summarizing, P = ( P, ≤ , ′ , ,
1) is an orthoposet.Finally, assume x ≤ y . Then, obviously, U ( y ) ⊆ U ( x, L ( x ′ y )). If, conversely, z ∈ U ( x, L ( x ′ , y )) then, since by (iii) of Lemma 4.1 we have( x ′ ⊓ t ) ⊓ (( x ⊔ y ) ⊓ t ) ∈ L ( x ′ , y )for all t ∈ P , we have( x ⊔ z ) ⊔ ((( x ′ ⊓ t ) ⊓ (( x ⊔ y ) ⊓ t )) ⊔ z ) = z ⊔ z = z for all t ∈ P whence by (i) z = ( x ⊔ y ) ⊔ z ∈ U ( x ⊔ y ) = U ( y ) . This shows U ( x, L ( x ′ , y )) ⊆ U ( y ), thus U ( x, L ( x ′ , y )) = U ( y ). Hence, P is a generalizedorthomodular poset.Consider the following identity:(i’) x ⊔ y ≤ ( x ⊔ z ) ⊔ (( x ′ ⊓ ( x ⊔ y )) ⊔ z ).Let • A denote the class of all algebras ( P, ⊔ , ′ , ,
1) of type (2 , , ,
0) satisfying identities(ii) – (iv) of Theorem 4.2 as well as condition (i) of Theorem 4.2, • W denote the variety of all algebras ( P, ⊔ , ′ , ,
1) of type (2 , , ,
0) satisfying iden-tities (ii) – (iv) of Theorem 4.2 as well as identity (i’).
Lemma 4.3.
The class W is a subvariety of the class A .Proof. Let A = ( P, ⊔ , ′ , ,
1) be a member of the variety W and assume that( x ⊔ z ) ⊔ ((( x ′ ⊓ w ) ⊓ (( x ⊔ y ) ⊓ w )) ⊔ z ) = z for all w ∈ P. Putting w = 1 we derive the identity( x ⊔ z ) ⊔ (( x ′ ⊓ ( x ⊔ y )) ⊔ z ) ≈ z. Hence x ⊔ y ≤ z follows by (i’), i.e. ( x ⊔ y ) ⊔ z = z . This shows that (i) holds and therefore A belongs to the class A .We are now interested in the congruence properties of the class A and the variety W (seee.g. [3]). For this, we recall the following concepts.Let C be a class of algebras of the same type and V a variety. Then the class C is called • congruence permutable if Θ ◦ Φ = Φ ◦ Θ for all A ∈ C and Θ , Φ ∈ Con A , • congruence distributive if (Θ ∨ Φ) ∧ Ψ = (Θ ∧ Ψ) ∧ (Φ ∧ Ψ) for all A ∈ C andΘ , Φ , Ψ ∈ Con A , 10 arithmetical if it is both congruence permutable and congruence distributive, • congruence regular if for each A = ( A, F ) ∈ C , every a ∈ A and all Θ , Φ ∈ Con A with [ a ]Θ = [ a ]Φ we have Θ = Φ.The following is well-known (cf. [3], Theorems 3.1.8, Corollary 3.2.4 and Theorem 6.1.3): • The class C is congruence permutable if there exists a so-called Maltsev term , i.e. aternary term p satisfying p ( x, x, y ) ≈ p ( y, x, x ) ≈ y, • The class C is congruence distributive if there exists a so-called majority term , i.e.a ternary term m satisfying m ( x, x, y ) ≈ m ( x, y, x ) ≈ m ( y, x, x ) ≈ x, • The variety V is congruence regular if and only if there exists a positive integer n and ternary terms t , . . . , t n such that t ( x, y, z ) = · · · = t n ( x, y, z ) = z if and only if x = y. Theorem 4.4.
The class A is congruence distributive and the variety W is arithmeticaland congruence regular.Proof. First consider class A . Define a ternary term m via m ( x, y, z ) := ( x ⊔ y ) ⊓ ( y ⊔ z ) ⊓ ( z ⊔ x ) . Then m ( x, x, z ) ≈ ( x ⊔ x ) ⊓ ( x ⊔ z ) ⊓ ( z ⊔ x ) ≈ x ⊓ ( x ⊔ z ) ≈ x,m ( x, y, x ) ≈ ( x ⊔ y ) ⊓ ( y ⊔ x ) ⊓ ( x ⊔ x ) ≈ ( x ⊔ y ) ⊓ x ≈ x,m ( x, z, z ) ≈ ( x ⊔ z ) ⊓ ( z ⊔ z ) ⊓ ( z ⊔ x ) ≈ ( x ⊔ z ) ⊓ z ≈ z proving that m is a majority term.Now consider variety W . We use the identity x ⊔ y ≤ x ⊔ ( x ′ ⊓ ( x ⊔ y ))which follows from (i’) by putting z = 0.Define a ternary term p via p ( x, y, z ) := ( x ⊔ ( y ′ ⊓ ( y ⊔ z ))) ⊓ ( z ⊔ ( y ′ ⊓ ( y ⊔ x ))) . Then p ( x, x, z ) ≈ ( x ⊔ ( x ′ ⊓ ( x ⊔ z ))) ⊓ ( z ⊔ ( x ′ ⊓ ( x ⊔ x ))) ≈ ( x ⊔ ( x ′ ⊓ ( x ⊔ z ))) ⊓ z ≈ z,p ( x, z, z ) ≈ ( x ⊔ ( z ′ ⊓ ( z ⊔ z ))) ⊓ ( z ⊔ ( z ′ ⊓ ( z ⊔ x ))) ≈ x ⊓ ( z ⊔ ( z ′ ⊓ ( z ⊔ x ))) ≈ x. Thus p is a Maltsev term. 11ow put t ( x, y ) := ( x ′ ⊓ ( x ⊔ y )) ⊔ ( y ′ ⊓ ( x ⊔ y )) . Then t ( x, x ) ≈ ( x ′ ⊓ ( x ⊔ x )) ⊔ ( x ′ ⊓ ( x ⊔ x )) ≈ , and if t ( x, y ) = 0 then x ′ ⊓ ( x ⊔ y ) = y ′ ⊓ ( x ⊔ y ) = 0 and hence x ≤ x ⊔ y ≤ x ⊔ ( x ′ ⊓ ( x ⊔ y )) = x ⊔ x,y ≤ x ⊔ y ≤ y ⊔ ( y ′ ⊓ ( x ⊔ y )) = y ⊔ y whence x = x ⊔ y = y . If, finally, t ( x, y, z ) := t ( x, y ) ⊔ z,t ( x, y, z ) := ( t ( x, y )) ′ ⊓ z then t ( x, x, z ) ≈ t ( x, x ) ⊔ z ≈ z,t ( x, x, z ) ≈ ( t ( x, x )) ′ ⊓ z ≈ z, and if t ( x, y, z ) = t ( x, y, z ) = z then t ( x, y ) ≤ z ≤ ( t ( x, y )) ′ and hence t ( x, y ) = t ( x, y ) ⊓ ( t ( x, y )) ′ = 0 whence x = y .It is a question whether the class A is congruence permutable, too. We can establish a“partial” Maltsev term as follows. If p ( x, y, z ) := ( x ∨ L ( y ′ , y ⊔ z )) ⊓ ( z ∨ L ( y ′ , x ⊔ y ))then p ( x, x, z ) = ( x ∨ L ( x ′ , x ⊔ z )) ⊓ ( z ∨ L ( x ′ , x ⊔ x )) = ( x ⊔ z ) ⊓ ( z ∨
0) = z,p ( x, z, z ) = ( x ∨ L ( z ′ , z ⊔ z )) ⊓ ( z ∨ L ( z ′ , x ⊔ z )) = ( x ∨ ⊓ ( x ⊔ z ) = x. The problem, however, is that within the “term” p there occurs the operator L and,moreover, the suprema occurring in p need not exist for all possible entries x, y, z , thusour “term” p is only partial. It is well-known that the Dedekind-MacNeille completion of an orthomodular poset neednot be an orthomodular lattice. The aim of this section is to show that a boundedposet with a unary operation is a strong generalized orthomodular poset if and only if itsDedekind-MacNeille completion is nearly an orthomodular lattice.The construction is as follows:Let P = ( P, ≤ , ′ , ,
1) be a bounded poset with a unary operation. DefineDM( P ) := { L ( A ) | A ⊆ P } ,A ∨ B := LU ( A, B ) for all
A, B ∈ DM( P ) ,A ∧ B := A ∩ B for all A, B ∈ DM( P ) ,A ∗ := L ( A ′ ) for all A ∈ DM( P ) , DM ( P ) := (DM( P ) , ∨ , ∧ , ∗ , { } , P ) . DM ( P ) is a complete lattice with a unary operation, called the Dedekind-MacNeillecompletion of P , and x L ( x ) is an isomorphism from ( P, ≤ , ′ ) to ( { L ( x ) | x ∈ P } , ⊆ , ∗ ).An orthomodular lattice is a generalized orthomodular poset which is a lattice, i.e. whichsatisfies the orthomodular law x ≤ y implies y = x ∨ ( x ′ ∧ y )or, equivalently, x ≤ y implies x = y ∧ ( x ∨ y ′ ) . DM ( P ) is said to be nearly an orthomodular lattice if L ( a ) ⊆ B implies B = L ( a ) ∨ ( B ∧ ( L ( a )) ∗ )for all a ∈ P and B ∈ DM( P ). Note that if DM ( P ) is an orthomodular lattice thenit is also nearly an orthomodular lattice, but our assumption is weaker since we do notquantify over all A, B ∈ DM( P ) with A ⊆ B , but only over all a ∈ P and all subsets B of DM( P ) with L ( a ) ⊆ B . Theorem 5.1.
Let P = ( P, ≤ , ′ , , be a bounded poset with a unary operation. Then P is a strong generalized orthomodular poset if and only if DM ( P ) is nearly an orthomodularlattice.Proof. It is easy to see that P is an orthoposet if and only if DM ( P ) is an ortholattice.Now let a, b ∈ P , A, B ∈ DM( P ) and C ⊆ P . Since L ( b ) ∧ ( A ∨ ( L ( b )) ∗ ) = L ( b ) ∩ LU ( A, L ( b ′ )) = L ( b, U ( A, b ′ ))the following are equivalent: P is a strong generalized orthomodular poset ,L ( C ) ≤ b implies L ( C ) = L ( b, U ( L ( C ) , b ′ )) ,A ⊆ L ( b ) implies A = L ( b, U ( A, b ′ )) ,A ⊆ L ( b ) implies A = L ( b ) ∧ ( A ∨ ( L ( b )) ∗ ) ,L ( b ′ ) ⊆ A ∗ implies A ∗ = L ( b ′ ) ∨ ( A ∗ ∧ L ( b )) ,L ( a ) ⊆ B implies B = L ( a ) ∨ ( B ∧ ( L ( a )) ∗ ) , DM ( P ) is nearly an orthomodular lattice . Corollary 5.2.
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