aa r X i v : . [ m a t h . L O ] J un Consistent posets
Ivan Chajda and Helmut L¨anger Abstract
We introduce so-called consistent posets which are bounded posets with anantitone involution ′ where the lower cones of x, x ′ and of y, y ′ coincide provided x, y are different form 0 , x, y are different form 0 then theirlower cone is different form 0, too. We show that these posets can be representedby means of commutative meet-directoids with an antitone involution satisfyingcertain identities and implications. In the case of a finite distributive or stronglymodular consistent poset, this poset can be converted into a residuated structureand hence it can serve as an algebraic semantics of a certain non-classical logicwith unsharp conjunction and implication. Finally we show that the Dedekind-MacNeille completion of a consistent poset is a consistent lattice, i.e. a boundedlattice with an antitone involution satisfying the above mentioned properties. AMS Subject Classification:
Keywords:
Consistent poset, antitone involution, distributive poset, strongly modularposet, commutative meet-directoid, residuation, adjointness, Dedekind-MacNeille com-pletion
In some non-classical logics the contraposition law is assumed. An algebraic semantics ofsuch logics is provided by means of De Morgan posets, i.e. bounded posets equipped witha unary operation ′ which is an antitone involution. This operation ′ is then consideredas a negation. Clearly, 0 ′ = 1 and 1 ′ = 0, but we do not ask ′ to be a complementation.In particular, this is the case of the logic of quantum mechanics represented by meansof an orthomodular lattice or an orhomodular poset in a broad sense. In orthomodularlattices the following implication holds x ≤ y and y ∧ x ′ = 0 imply x = y. In fact, for an ortholattice this condition is necessary and sufficient for being orthomod-ular. When working with orthomodular posets, the aforementioned condition can beexpressed in the form x ≤ y and L ( y, x ′ ) = { } imply x = y Corresponding author 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. L ( y, x ′ ) denotes the lower cone of y and x ′ .However, there are logics where such a condition can be recognized as too restrictive.Hence, we can relax the equality x = y by asking that x, y have the same lower conesgenerated by the pairs including the involutive members, i.e. we consider the condition x ≤ y and L ( y, x ′ ) = { } imply L ( x, x ′ ) = L ( y, y ′ ) . Of course, if P = ( P, ≤ , ′ , ,
1) is a bounded poset where the operation ′ is a complemen-tation then L ( x, x ′ ) = { } = L ( y, y ′ )for all x, y ∈ P . However, this is rather restrictive. Hence, we do not ask in general that ′ is a complementation, but P should satisfy L ( x, x ′ ) = L ( y, y ′ ) for x, y = 0 , x, y are comparable but,on the other hand, we will ask that L ( x, y ) = { } if and only if at least one of the entries x, y is equal to 0. Such a poset will be called consistent in the sequel. It representscertain logics satisfying De Morgan’s laws. Usually, a logic is considered to be well-founded if it contains a logical connective implication which is related with conjunctionvia the so-called adjointness. In what follows, we show that consistent posets can berepresented by means of algebras (with everywhere defined operations) which enables touse algebraic tools for investigating these posets. Moreover, we show when these posetscan be organized into a kind of residuated structure, i.e. we introduce conjunction andimplication related via adjointness. Of course, working with posets, one cannot expectthat these logical connectives will be operations giving a unique result for given entries.We will define operators assigning to the couple x, y of entries a certain subset of P . Itis in accordance with the description of uncertainty of such a logic based on the fact thata poset instead of a lattice is used. In our previous papers [4] and [6] we studied complemented posets. We showed whensuch a poset can be represented by a commutative directoid ([1], [3] and [10]) and whenit can be organized into a residuated or left-residuated structure ([2], [4], [5], [6] and [8]).Now we introduce a bit more general posets with an antitone involution which need notbe a complementation but it still shares similar properties. We again try to characterizethese posets by identities or implications of corresponding commutative meet-directoidssimilarly as it was done in [1]. This approach has the advantage that commutativedirectoids are algebras similar to semilattices and hence we can use standard algebraictools for their constructions, see e.g. [10]. We also solve the problem when these so-calledconsistent posets can be converted into residuated or left-residuated structures.For the reader’s convenience, we recall several concepts concerning posets.Let P = ( P, ≤ ) be a poset, a, b ∈ P and A, B ⊆ P . We write a k b if a and b areincomparable and we extend ≤ to subsets by defining A ≤ B if and only if x ≤ y for all x ∈ A and y ∈ B. { a } ≤ B and A ≤ { b } we also write a ≤ B and A ≤ b , respectively. Analogousnotations are used for the reverse order ≥ . Moreover, we define L ( A ) := { x ∈ P | x ≤ A } ,U ( A ) := { x ∈ P | A ≤ x } . Instead of L ( A ∪ B ), L ( { a } ∪ B ), L ( A ∪ { b } ) and L ( { a, b } ) we also write L ( A, B ), L ( a, B ), L ( A, b ) and L ( a, b ), respectively. Analogous notations are used for U . Instead of L ( U ( A ))we also write LU ( A ). Analogously, we proceed in similar cases. Sometimes we identifysingletons with their unique element, so we often write L ( a, b ) = 0 and U ( a, b ) = 1 insteadof L ( a, b ) = { } and U ( a, b ) = { } , respectively. The poset P is called downward directed if L ( x, y ) = ∅ for all x, y ∈ P . Of course, every poset with 0 is downward directed. The poset P is called bounded if it has a least element 0 and a greatest element 1. This factwill be expressed by notation ( P, ≤ , , P is called modular if(1) x ≤ z implies L ( U ( x, y ) , z ) = LU ( x, L ( y, z )).This is equivalent to x ≤ z implies U L ( U ( x, y ) , z ) = U ( x, L ( y, z )) . Recall from [7] that P is called strongly modular if it satisfies the LU-identities(2) L ( U ( x, y ) , U ( x, z )) ≈ LU ( x, L ( y, U ( x, z ))),(3) L ( U ( L ( x, z ) , y ) , z ) ≈ LU ( L ( x, z ) , L ( y, z )).These are equivalent to U L ( U ( x, y ) , U ( x, z )) ≈ U ( x, L ( y, U ( x, z ))) ,U L ( U ( L ( x, z ) , y ) , z ) ≈ U ( L ( x, z ) , L ( y, z )) , respectively. Observe that in case x ≤ z both (2) and (3) yield (1). Hence, every stronglymodular poset is modular. Moreover, every modular lattice is a strongly modular poset.A strongly modular poset which is not a lattice is presented in Example 3.4.The poset P is called distributive if it satisfies the following identity:(4) L ( U ( x, y ) , z ) ≈ LU ( L ( x, z ) , L ( y, z )).This identity is equivalent to every single one of the following identities (see [11]): U L ( U ( x, y ) , z ) ≈ U ( L ( x, z ) , L ( y, z )) ,U ( L ( x, y ) , z ) ≈ U L ( U ( x, z ) , U ( y, z )) ,LU ( L ( x, y ) , z ) ≈ L ( U ( x, z ) , U ( y, z )) . In fact, the inclusions LU ( L ( x, z ) , L ( y, z )) ⊆ L ( U ( x, y ) , z ) ,U L ( U ( x, z ) , U ( y, z )) ⊆ U ( L ( x, y ) , z )hold in every poset. Hence, to check distributivity, we need only to confirm one of theconverse inclusions. Observe that in case x ≤ z (4) implies (1). Hence every distributiveposet is modular. Distributivity does not imply strong modularity. A unary operation ′ on P is called 3 antitone if, for all x, y ∈ P , x ≤ y implies y ′ ≤ x ′ , • an involution if it satisfies the identity x ′′ ≈ x , • a complementation if L ( x, x ′ ) ≈ U ( x, x ′ ) ≈ poset is called Boolean if it is distributive and has a unary operation which is a com-plementation. For A ⊆ P we definemax A := set of all maximal elements of A, max A := set of all minimal elements of A,A ′ := { x ′ | x ∈ A } . If the poset is bounded and distributive, we can prove the following property of an antitoneinvolution.
Lemma 2.1.
Let ( P, ≤ , ′ , , be a bounded distributive poset with an antitone involutionand a, b ∈ P with a ≤ b and L ( b, a ′ ) = { } . Then the following hold: L ( a, a ′ ) = L ( b, b ′ ) = { } ,U ( a, a ′ ) = U ( b, b ′ ) = { } . Proof.
We have L ( a, a ′ ) = LU L ( a, a ′ ) = LU ( L ( a, a ′ ) ,
0) = LU ( L ( a, a ′ ) , L ( b, a ′ )) = L ( U ( a, b ) , a ′ ) == L ( U ( b ) , a ′ ) = L ( b, a ′ ) = { } ,L ( b, b ′ ) = LU L ( b ′ , b ) = LU (0 , L ( b ′ , b )) = LU ( L ( a ′ , b ) , L ( b ′ , b )) = L ( U ( a ′ , b ′ ) , b ) == L ( U ( a ′ ) , b ) = L ( a ′ , b ) = { } ,U ( a, a ′ ) = ( L ( a ′ , a )) ′ = { } ′ = { } ,U ( b, b ′ ) = ( L ( b ′ , b )) ′ = { } ′ = { } . Now we recall the concept of a commutative meet-directoid from [10], see also [3] for de-tails. We will use it for the characterization of consistent posets which will be introducedbelow. The advantage of this approach is that we characterize properties of posets bymeans of identities and quasiidentities of algebras. Hence, one can use algebraic tools fortheir investigation.A commutative meet-directoid (see [3] and [10]) is a groupoid D = ( D, ⊓ ) satisfying thefollowing identities: x ⊓ x ≈ x (idempotency) ,x ⊓ y ≈ y ⊓ x (commutativity) , ( x ⊓ ( y ⊓ z )) ⊓ z ≈ x ⊓ ( y ⊓ z ) (weak associativity) . Let P = ( P, ≤ ) be a downward directed poset. Define x ⊓ y := x ∧ y for comparable x, y ∈ P and let x ⊓ y = y ⊓ x be an arbitrary element of L ( x, y ) if x, y ∈ P areincomparable. Then D ( P ) := ( P, ⊓ ) is a commutative meet-directoid which is called a meet-directoid assigned to P . Conversely, if D = ( D, ⊓ ) is a commutative meet-directoidand we define for all x, y ∈ D x ≤ y if and only if x ⊓ y = x then P ( D ) := ( D, ≤ ) is a downward directed poset, the so-called poset induced by D .Though the assignment P D ( P ) is not unique, we have P ( D ( P )) = P for every down-ward directed poset P . Sometimes we consider posets and commutative meet-directoidstogether with a unary operation. Let ( D, ⊓ , ′ ) be a commutative meet-directoid ( D, ⊓ , ′ )with an antitone involution, i.e. ′ is antitone with respect to the partial order relationinduced by (5). We define x ⊔ y := ( x ′ ⊓ y ′ ) ′ for all x, y ∈ D. Then ⊔ is also idempotent, commutative and weakly associative and we have for all x, y ∈ D x ⊔ y = x ∨ y if x, y are comparable,x ⊔ y = y ⊔ x ∈ U ( x, y ) if x k y,x ⊓ y = x if and only if x ⊔ y = y,L ( x ) = { z ⊓ x | z ∈ P } ,U ( x ) = { z ⊔ x | z ∈ P } ,L ( x, y ) = { ( z ⊓ x ) ⊓ ( z ⊓ y ) | z ∈ P } ,U ( x, y ) = { ( z ⊔ x ) ⊓ ( z ⊔ y ) | z ∈ P } . Posets with an antitone involution can be characterized in the language of commutativemeet-directoids by identities as follows. The following lemma was proved in [1]. For theconvenience of the reader we provide a proof.
Lemma 2.2.
Let P = ( P, ≤ , ′ ) be a downward directed poset with a unary operation and D ( P ) an assigned meet-directoid. Then P is a poset with an antitone involution if andonly if D ( P ) satisfies the identities (6) x ′′ ≈ x , (7) ( x ⊓ y ) ′ ⊓ y ′ ≈ y ′ .Proof. Condition (6) is evident by definition. Let a, b ∈ P . If (7) holds and a ≤ b then b ′ = ( a ⊓ b ) ′ ⊓ b ′ = a ′ ⊓ b ′ ≤ a ′ which shows that ′ is antitone. If, conversely, ′ is antitonethen from a ⊓ b ≤ b we obtain b ′ ≤ ( a ⊓ b ) ′ , i.e. ( a ⊓ b ) ′ ⊓ b ′ = b ′ which is (7). Now we define our key concept.
Definition 3.1. A consistent poset is a bounded poset ( P, ≤ , ′ , , with an antitoneinvolution satisfying the following two conditions: (8) L ( x, x ′ ) = L ( y, y ′ ) for all x, y ∈ P \ { , } , (9) L ( x, y ) = 0 for all x, y ∈ P \ { } .
5t is easy to see that an at least three-element bounded poset P = ( P, ≤ , ′ , ,
1) withan antitone involution is consistent if and only if P has exactly one atom a such that P = [ a, a ′ ] ∪ { , } and ′ is a complementation on the interval ([ a, a ′ ] , ≤ ). Lemma 3.2.
The conditions (8) and (9) are independent.Proof.
The four-element Boolean algebra satisfies (8) but not (9), and the five-elementchain (together with its unique possible antitone involution) satisfies (9) but not (8).In the following we list examples of consistent posets.
Example 3.3.
The poset depicted in Figure 1 ✉✉✉ ✉ ✉ ✉✉ ✉ ✉ ✉✉✉ ◗◗◗◗◗◗ ❆❆❆❆ ✁✁✁✁✑✑✑✑✑✑✑✑✑✑✑✑ ✁✁✁✁ ❆❆❆❆◗◗◗◗◗◗(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ ab c d ee ′ d ′ c ′ b ′ a ′ ′ Fig. 1 is consistent, but neither modular since L ( U ( b, d ) , e ′ ) = L ( a ′ , e ′ ) = L ( e ′ ) = L ( b ) = LU ( b ) = LU ( a, b ) = LU ( b, L ( d, e ′ )) , nor a lattice since d ′ and e ′ are different minimal upper bounds of b and c . Example 3.4.
The poset visualized in Figure 2 ✉✉ ✉ ✉ ✉✉ ✉ ✉ ✉✉✉ ◗◗◗◗◗◗ ❆❆❆❆ ✁✁✁✁✑✑✑✑✑✑✑✑✑✑✑✑ ✁✁✁✁ ❆❆❆❆◗◗◗◗◗◗(cid:0)(cid:0)(cid:0)(cid:0)✟✟✟✟✟✟✟✟❅❅❅❅ ✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍ (cid:0)(cid:0)(cid:0)(cid:0)❍❍❍❍❍❍❍❍ ❅❅❅❅ ab c d ee ′ d ′ c ′ b ′ a ′ ′ Fig. 2 is consistent and strongly modular, but not a lattice since b ′ and e ′ are different minimalupper bounds of c and d . Example 3.5.
The poset depicted in Figure 3 ✉✉✉ ✉ ✉ ✉✉ ✉✉ ✉ ✉ ✉✉✉ ◗◗◗◗◗◗ ❆❆❆❆ ✁✁✁✁✑✑✑✑✑✑✑✑✑✑✑✑ ✁✁✁✁ ❆❆❆❆◗◗◗◗◗◗(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ ❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅ ❅❅❅❅ ab c d ef f ′ d ′ e ′ c ′ b ′ a ′ ′ Fig. 3 is consistent and distributive, but neither Boolean since L ( a, a ′ ) = a = 0 , nor a latticesince c ′ and d ′ are different minimal bounds of b and e . Using the language of commutative meet-directoids, we can easily characterize lower cones L ( a, b ) as follows. 7 emma 3.6. Let ( P, ≤ ) be a downward directed poset, a, b, c ∈ P and ( P, ⊓ ) an assignedmeet-directoid. Then c ∈ L ( a, b ) if and only if c = ( c ⊓ a ) ⊓ ( c ⊓ b ) .Proof. If c ∈ L ( a, b ) then c = c ⊓ c = ( c ⊓ a ) ⊓ ( c ⊓ b ). If, conversely, c = ( c ⊓ a ) ⊓ ( c ⊓ b )then c ≤ c ⊓ a ≤ a,c ≤ c ⊓ b ≤ b and hence c ∈ L ( a, b ).Now we characterize consistent posets by means of commutative meet-directoids. Theorem 3.7.
Let P = ( P, ≤ , ′ , , be a bounded poset with a unary operation and D ( P ) an assigned meet-directoid. Then P is consistent if and only if D ( P ) satisfies identities (6) and (7) and implications (10) and (11) : (10) x, y = 0 , and z = ( z ⊓ x ) ⊓ ( z ⊓ x ′ ) imply z = ( z ⊓ y ) ⊓ ( z ⊓ y ′ ) , (11) if z = ( z ⊓ x ) ⊓ ( z ⊓ y ) implies z = 0 then x = 0 or y = 0 .Proof. (10) According to Lemma 3.6 the following are equivalent:(10) , if x, y = 0 , z ∈ L ( x, x ′ ) then z ∈ L ( y, y ′ ) , if x, y = 0 , L ( x, x ′ ) ⊆ L ( y, y ′ ) , if x, y = 0 , L ( x, x ′ ) = L ( y, y ′ ) . (11) According to Lemma 3.6 the following are equivalent:(11) , if x, y = 0 then there exists some z = 0 with z ∈ L ( x, y ) , if x, y = 0 then L ( x, y ) = 0 . Lemma 2.2 completes the proof.We can also characterize downward directed distributive posets in a similar manner. Thefollowing theorem was proved in [9]. For the convenience of the reader we provide a proof.
Theorem 3.8.
Let P = ( P, ≤ ) be a downward directed poset and D ( P ) an assignedmeet-directoid. Then P is distributive if and only if D ( P ) satisfies implication (12) : (12) w ⊓ (( t ⊔ x ) ⊔ ( t ⊔ y )) = w ⊓ z = w and s ⊔ (( t ⊓ x ) ⊓ ( t ⊓ z )) = s ⊔ (( t ⊓ y ) ⊓ ( t ⊓ z )) = s for all t ∈ P imply w ≤ s . roof. Since U ( x, y ) = { ( t ⊔ x ) ⊔ ( t ⊔ y ) | t ∈ P } ,w ⊓ u = w is equivalent to w ∈ L ( u ) ,w ⊓ (( t ⊔ x ) ⊔ ( t ⊔ y )) = w ⊓ z = w is equivalent to w ∈ L ( U ( x, y ) , z ). Further, since L ( x, z ) = { ( t ⊓ x ) ⊓ ( t ⊓ z ) | t ∈ P } ,L ( y, z ) = { ( t ⊓ y ) ⊓ ( t ⊓ z ) | t ∈ P } ,s ⊔ u = s is equivalent to s ∈ U ( u ) ,s ⊔ (( t ⊓ x ) ⊓ ( t ⊓ z )) = s ⊔ (( t ⊓ y ) ⊓ ( t ⊓ z )) = s is equivalent to s ∈ U ( L ( x, z ) , L ( y, z )).Hence the following are equivalent:(12) ,w ∈ L ( U ( x, y ) , z ) and s ∈ U ( L ( x, z ) , L ( y, z )) imply w ≤ s,L ( U ( x, y ) , z ) ⊆ LU ( L ( x, z ) , L ( y, z )) , P is distributive . Definition 4.1. A consistent residuated poset is an ordered six-tuple ( P, ≤ , ⊙ , → , , where ( P, ≤ , , is a bounded poset and ⊙ and → are mappings ( so-called operators ) from P to P satisfying the following conditions for all x, y, z ∈ P : • x ⊙ y ≈ y ⊙ x , • x ⊙ ≈ ⊙ x ≈ x , • x ⊙ y ≤ z if and only if x ≤ y → z ( adjointness ) . Let ( P, ≤ , ′ , ,
1) be a poset with an antitone involution. Define mappings ⊙ and → from P to 2 P as follows:(13) x ⊙ y := (cid:26) x ≤ y ′ , max L ( x, y ) otherwise x → y := (cid:26) x ≤ y, min U ( x ′ , y ) otherwise Theorem 4.2.
Let ( P, ≤ , ′ , , be a finite distributive consistent poset and ⊙ and → bedefined by (13) . Then ( P, ≤ , ⊙ , → , , is a consistent residuated poset.Proof. Let a, b, c ∈ P . Because a ≤ b ′ is equivalent to b ≤ a ′ and, moreover, L ( a, b ) = L ( b, a ), ⊙ is commutative. Further,if a = 0 then a ⊙ a, if a = 0 then a ⊙ L ( a,
1) = max L ( a ) = a. By commutativity of ⊙ we obtain a ⊙ ⊙ a = a . We consider the following cases:9 a ≤ b ′ and b ≤ c .Then a ⊙ b = 0 ≤ c and a ≤ b → c . • a ≤ b ′ and b c .Then a ⊙ b = 0 ≤ c and a ≤ b ′ ≤ min U ( b ′ , c ) = b → c . • a b ′ and b ≤ c .Then a ⊙ b = max L ( a, b ) ≤ b ≤ c and a ≤ b → c . • a b ′ , b c .In case a = 1, a ⊙ b ≤ c and a ≤ b → c are not possible because a ⊙ b = 1 ⊙ b = b c .Moreover, b, c ′ = 0 and therefore b → c = min U ( b ′ , c ) = (max L ( b, c ′ )) ′ = 0 ′ = 1whence a = 1 b → c .Similarly, in case c = 0, a ⊙ b ≤ c and a ≤ b → c are not possible because a, b = 0 and therefore a ⊙ b = max L ( a, b ) = 0 whence a ⊙ b c . Moreover, a b ′ = min U ( b ′ ) = min U ( b ′ , c ) = b → c .In case b = 1 the following are equivalent: a ⊙ b ≤ c,a ⊙ ≤ c,a ≤ c,a ≤ min U ( c ) ,a ≤ min U (1 ′ , c ) ,a ≤ → c,a ≤ b → c. There remains the case a, b = 1 and c = 0. Then a, b, c = 0 ,
1. If a ⊙ b ≤ c thenmax L ( a, b ) ≤ c and hence L ( a, b ) ≤ c whence b → c = min U ( b ′ , c ) ⊆ U ( b ′ , c ) ⊆ U ( b ′ , a ⊙ b ) = U ( b ′ , L ( a, b )) == U L ( U ( b ′ , a ) , U ( b ′ , b )) = U L ( U ( b ′ , a ) , U ( a ′ , a )) ⊆ U LU ( a ) = U ( a )which implies a ≤ b → c . If, conversely, a ≤ b → c then a ≤ min U ( b ′ , c ) and hence a ≤ U ( b ′ , c ) whence a ⊙ b = max L ( a, b ) ⊆ L ( a, b ) ⊆ L ( b → c, b ) = L ( U ( b ′ , c ) , b ) == LU ( L ( b ′ , b ) , L ( c, b )) = LU ( L ( c ′ , c ) , L ( c, b )) ⊆ LU L ( c ) = L ( c )and hence a ⊙ b ≤ c .This shows that in any case a ⊙ b ≤ c is equivalent to a ≤ b → c .We now study residuation in not necessarily distributive consistent posets. For thispurpose, we slightly modify our definition of residuation by deleting the assumption ofcommutativity of ⊙ . Definition 4.3. A weak consistent residuated poset is an ordered six-tuple ( P, ≤ , ⊙ , → , , where ( P, ≤ , , is a bounded poset and ⊙ and → are mappings ( so-called opera-tors ) from P to P satisfying the following conditions for all x, y, z ∈ P : x ⊙ ≈ ⊙ x ≈ x , • x ⊙ y ≤ z if and only if x ≤ y → z ( adjointness ) . Let ( P, ≤ , ′ , ,
1) be a poset with an antitone involution. We modify the definition of themappings (so-called operators) ⊙ and → from P to 2 P in the following way:(14) x ⊙ y := (cid:26) x ≤ y ′ , max L ( U ( x, y ′ ) , y ) otherwise x → y := (cid:26) x ≤ y, min U ( x ′ , L ( x, y )) otherwiseNow, we are able to prove our second result on residuation. Theorem 4.4.
Let ( P, ≤ , ′ , , be a finite strongly modular consistent poset and ⊙ and → be defined by (14) . Then ( P, ≤ , ⊙ , → , , is a weak consistent residuated poset.Proof. Let a, b, c ∈ P . If a = 0 then a ⊙ a and 1 ⊙ a = 0 = a . If a = 0 then a ⊙ L ( U ( a, ′ ) ,
1) = max LU ( a ) = max L ( a ) = a, ⊙ a = max L ( U (1 , a ′ ) , a ) = max L ( a ) = a. We consider the following cases: • a ≤ b ′ and b ≤ c .Then a ⊙ b = 0 ≤ c and a ≤ b → c . • a ≤ b ′ and b c .Then a ⊙ b = 0 ≤ c and a ≤ b ′ ≤ min U ( b ′ , L ( b, c )) = b → c . • a b ′ and b ≤ c .Then a ⊙ b = max L ( U ( a, b ′ ) , b ) ≤ b ≤ c and a ≤ b → c . • a b ′ , b c .In case a = 1, a ⊙ b ≤ c and a ≤ b → c are not possible because a ⊙ b = 1 ⊙ b = b c .Moreover, b, c ′ = 0 and hence L ( b, c ′ ) = 0 which implies L ( b, U ( b ′ , c ′ )) = 0 andtherefore b → c = min U ( b ′ , L ( b, c )) = (max L ( b, U ( b ′ , c ′ ))) ′ = 0 ′ = 1 whence a = 1 b → c .Similarly, in case c = 0, a ⊙ b ≤ c and a ≤ b → c are not possible because a, b = 0 and hence L ( a, b ) = 0 whence L ( U ( a, b ′ ) , b ) = 0 and therefore a ⊙ b =max L ( U ( a, b ′ ) , b ) = 0 whence a ⊙ b c . Moreover, a b ′ = min U ( b ′ ) =min U ( b ′ , L ( b, c )) = b → c .In case b = 1 the following are equivalent: a ⊙ b ≤ c,a ⊙ ≤ c,a ≤ c,a ≤ min U ( c ) ,a ≤ min U L ( c ) ,a ≤ min U (1 ′ , L (1 , c )) ,a ≤ → c,a ≤ b → c. a, b = 1 and c = 0. Then a, b, c = 0 ,
1. If a ⊙ b ≤ c then b → c = min U ( b ′ , L ( b, c )) ⊆ U ( b ′ , L ( b, c )) ⊆ U ( b ′ , L ( b, a ⊙ b )) == U ( b ′ , L ( b, max L ( U ( a, b ′ ) , b ))) = U ( b ′ , L ( b ) ∩ L (max L ( U ( a, b ′ ) , b ))) == U ( b ′ , L ( b ) ∩ L ( U ( a, b ′ ) , b )) = U ( b ′ , L ( b, U ( a, b ′ ))) = U L ( U ( b ′ , b ) , U ( a, b ′ )) == U L ( U ( a ′ , a ) , U ( a, b ′ )) ⊆ U LU ( a ) = U ( a )which implies a ≤ b → c . If, conversely, a ≤ b → c then a ⊙ b = max L ( U ( a, b ′ ) , b ) ⊆ L ( U ( a, b ′ ) , b ) ⊆ L ( U ( b → c, b ′ ) , b ) == L ( U (min U ( b ′ , L ( b, c )) , b ′ ) , b ) = L ( U (min U ( b ′ , L ( b, c ))) ∩ U ( b ′ ) , b ) == L ( U ( b ′ , L ( b, c )) ∩ U ( b ′ ) , b ) = L ( U ( b ′ , L ( b, c )) , b ) = L ( U ( L ( b, c ) , b ′ ) , b ) == LU ( L ( b, c ) , L ( b ′ , b )) = LU ( L ( b, c ) , L ( c ′ , c )) ⊆ LU L ( c ) = L ( c )and hence a ⊙ b ≤ c .This shows that in any case a ⊙ b ≤ c is equivalent to a ≤ b → c . In what follows we investigate the question for which posets P with an antitone involutiontheir Dedekind-MacNeille completion DM ( P ) is a consistent lattice. A bounded lattice ( L, ∨ , ∧ , ′ , ,
1) with an antitone involution is called consistent if it is consistent whenconsidered as a poset, i.e. if x ∧ x ′ = y ∧ y ′ for all x, y ∈ L \ { , } ,x ∧ y = 0 for all x, y ∈ L \ { } . Let P = ( P, ≤ , ′ ) be a poset with an antitone involution. DefineDM( P ) := { L ( A ) | A ⊆ P } ,A ∗ := L ( A ′ ) for all A ∈ DM( P ) , DM ( P ) := (DM( P ) , ⊆ , ∗ )Then DM ( P ) is a complete lattice with an antitone involution, called the Dedekind-MacNeille completion of P . That ∗ is an antitone involution on (DM( P ) , ⊆ ) can be seen asfollows. Let A, B ∈ DM( P ). If A ⊆ B then A ′ ⊆ B ′ and hence B ∗ = L ( B ′ ) ⊆ L ( A ′ ) = A ∗ .Moreover, A ∗∗ = L (( L ( A ′ )) ′ ) = LU ( A ) = A . We have( L ( A )) ∗ = L (( L ( A )) ′ ) = LU ( A ′ ) for all A ⊆ P,A ∨ B = LU ( A, B ) for all
A, B ∈ DM( P ) ,A ∧ B = A ∩ B for all A, B ∈ DM( P ) . Theorem 5.1.
Let P = ( P, ≤ , ′ ) be a poset with an antitone involution. Then DM ( P ) is a consistent lattice if and only if P is a consistent poset. roof. Assume P to be a consistent poset. Further assume A ⊆ P and L ( A ) = 0 , P . Then1 / ∈ L ( A ) and there exists some a ∈ L ( A ) \ { } . Hence 0 / ∈ U ( A ′ ) and a ′ ∈ U ( A ′ ) \ { } .Now L ( A ) ∧ ( L ( A )) ∗ = L ( A ) ∩ LU ( A ′ ) = [ x ∈ L ( A ) L ( x ) ∩ \ y ∈ U ( A ′ ) L ( y ) == [ x ∈ L ( A ) \{ } L ( x ) ∩ \ y ∈ U ( A ′ ) \{ } L ( y ) . Now L ( a, a ′ ) = \ y ∈ U ( A ′ ) \{ } L ( y ′ , y ) = \ y ∈ U ( A ′ ) \{ } ( L ( y ′ ) ∩ L ( y )) ⊆⊆ \ y ∈ U ( A ′ ) \{ } ( [ x ∈ L ( A ) \{ } L ( x ) ∩ L ( y )) = [ x ∈ L ( A ) \{ } L ( x ) ∩ \ y ∈ U ( A ′ ) \{ } L ( y ) == [ x ∈ L ( A ) \{ } ( L ( x ) ∩ \ y ∈ U ( A ′ ) \{ } L ( y )) ⊆ [ x ∈ L ( A ) \{ } ( L ( x ) ∩ L ( x ′ )) == [ x ∈ L ( A ) \{ } L ( x, x ′ ) = L ( a, a ′ )and hence L ( A ) ∧ ( L ( A )) ∗ = L ( a, a ′ ). This shows L ( A ) ∧ ( L ( A )) ∗ = L ( B ) ∧ ( L ( B )) ∗ for all A, B ⊆ P with L ( A ) , L ( B ) = 0 , P . Now assume A, B ⊆ P and L ( A ) , L ( B ) = 0.Then there exists some a ∈ L ( A ) \ { } and some b ∈ L ( B ) \ { } . Since P is consistentthere exists some c ∈ L ( a, b ) \ { } . Now L ( c ) ⊆ L ( a ) ⊆ L ( A ), L ( c ) ⊆ L ( b ) ⊆ L ( B )and 0 = c ∈ L ( c ) and hence L ( c ) = 0. This shows that DM ( P ) is a consistent latticeprovided P is a consistent poset. The converse is evident.Compliance with Ethical Standards: This study was funded by the Austrian ScienceFund (FWF), project I 4579-N, and the Czech Science Foundation (GA ˇCR), project 20-09869L, as well as by ¨OAD, project CZ 02/2019, and, concerning the first author, byIGA, project PˇrF 2020 014. The authors declare that they have no conflict of interest.This article does not contain any studies with human participants or animals performedby any of the authors. References [1] I. Chajda, M. Kolaˇr´ık and H. L¨anger, Varieties corresponding to classes of comple-mented posets. Miskolc Math. Notes (submitted). http://arxiv.org/abs/1911.05138.[2] I. Chajda, M. Kolaˇr´ık and H. L¨anger, Extensions of posets with an an-titone involution to residuated structures. Fuzzy Sets Systems (submitted).http://arxiv.org/abs/2004.14127.[3] I. Chajda and H. L¨anger, Directoids. An Algebraic Approach to Ordered Sets. Hel-dermann, Lemgo 2011. ISBN 978-3-88538-232-4.[4] I. Chajda and H. L¨anger, Orthomodular posets can be organized as conditionallyresiduated structures. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. (2014),29–33. 135] I. Chajda and H. L¨anger, Residuation in orthomodular lattices. Topological AlgebraAppl. (2017), 1–5.[6] I. Chajda and H. L¨anger, Residuated operators in complemented posets. Asian-European J. Math. (2018), 1850097 (15 pages).[7] I. Chajda and H. L¨anger, Residuation in modular lattices and posets. Asian-European J. Math. (2019), 1950092 (10 pages).[8] I. Chajda and H. L¨anger, Residuation in finite posets. Math. Slovaca (submitted).http://arxiv.org/abs/1910.09009.[9] I. Chajda and H. L¨anger, Kleene posets and pseudo-Kleene posets. Order (submit-ted). http://arxiv.org/abs/2006.04417.[10] J. Jeˇzek and R. Quackenbush, Directoids: algebraic models of up-directed sets. Al-gebra Universalis (1990), 49–69.[11] J. Larmerov´a and J. Rach˚unek, Translations of distributive and modular orderedsets. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math.27