Hilbert Algebras As Implicative Partial Semilattices

Home , Dual (category theory), Heyting algebra

DOI: 10.2478/s11533-007-0008-2 Research article CEJM 5(2) 2007 264–279

Hilbert algebras as implicative partial semilattices

J¯anis C¯ırulis∗

University of Latvia, LV-1586 Riga, Latvia

Received 15 August 2006; accepted 16 February 2007

Abstract: The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra iff both algebras have the same filters. An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice), and conversely. The implication in an implicative partial semilattice is characterised in terms of filters of the underlying partial semilattice. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.

Keywords: Compatible elements, ﬁlter, Hilbert algebra, implication, implicative semilattice, meet, partial semilattice MSC (2000): 06A12, 03G25

1 Introduction

Hilbert algebras, or positive implication algebras [35], are the duals of Henkin algebras called by him implicative models in [21] (this term has also been used for Hilbert algebras [30, 31]). Positive implicative BCK-algebras [24] are actually another version of Henkin algebras (see, e.g. [28, 37]). As a matter of fact, these algebras are an algebraic counterpart of positive implica- tional calculus. Various expansions of Hilbert algebras by a conjunction-like operation have also been studied in the literature. The most extensively investigated among them are implicative semilattices [13, 34], which are known also as Brouwerian semilattices. The so called (H)-Hilbert algebras [22] also have turned out to be another form of implicative semilattices. ∗ E-mail: [email protected] J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 265

Hilbert algebras in which all meets exist also have been studied; an important recent paper on this subject is [16]. Unfortunately, both these algebras and implicative semilattices have been mentioned in the literature under the same name Hertz algebras; this has caused several misunderstandings – see Section 3 below for a brief account of them; cf. also the discussion in [10, 17]. It was demonstrated in [12] by the present author that certain constructions and results can be transferred from implicative semilattices to Hilbert algebras equipped with the so called compatible meet operation. The infimum of elements a and b of a Hilbert algebra is said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense that goes back to [30]. Therefore, this operation is normally partial. In this paper we study elementary algebraic properties of the compatible meet operation and its connection with implication. In particular, we show in what sense Hilbert algebras can be treated as implicative partial semilattices. The paper is structured as follows. Section 2 contains general information on Hilbert algebras, including a description of their filter lattices. Section 3 is concerned with compatible meets and partial semilattices. It is shown here that a Hilbert algebra is a partial semilattice in the sense of [18] with respect to the compatible meet operation, and that a partial semilattice is a reduct of a Hilbert algebra if and only if both algebras have the same filters. The concept of an implicative partial semilattice as a relative subalgebra of an implicative semilattice is introduced in Section 4. Also a theorem stating that implicative partial semilattices are just Hilbert algebras equipped with the partial compatible meet operation is proved here. Furthermore, implication in partial semilattices is characterized both by some explicit conditions on this operation and in terms of filters.

2 Preliminaries: Hilbert algebras

An implicative algebra (see [35]) may be deﬁned as an algebra (A, →, 1) of type (2,0) such that the relation introduced on A by

x ≤ y if and only if x → y =1 (1) is a partial order with 1 the greatest element. A positive implication algebra [35]or, more concisely, Hilbert algebra [14, 15] is an implicative algebra in which the following inequalities hold: →1 : x ≤ y → x, →2 : x → .y → z ≤ x → y. → .x → z. Following [9, 12], we avoid parentheses as a tool for grouping in favour of dots. For example, the term (x → ((y → z) → x)) → (z → (x → y)) is coded in the dot notation as x → :y → z. → x∴ → :z → .x → y. The above characterization of Hilbert algebras is a bit redundant. On the other hand, the class of all Hilbert algebras considered as algebras of the form (A, →, 1) is equationally deﬁnable [14, 15]. The following useful algebraic properties of the operation → in Hilbert algebras can be found, e.g., in [4, 12, 14, 15, 31, 35]. 266 J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279

→3 : x ≤ 1, →4 : x → 1=1, →5 : x → x =1, →6 :ifx ≤ y,theny → z ≤ x → z, →7 :ifx ≤ y,thenz → x ≤ z → y, →8 : x → .x → y = x → y, →9 : x → .y → z = y → .x → z, →10 : x → .y → z = x → y. → .x → z, →11 : x → y. → y: → y = x → y, →12 : x ≤ x → y. → y. A Hilbert algebra A is said to be an implication algebra [35] if the inequality →13 : x → y. → x ≤ x or, equivalently, if any of the equations →14 : x → y. → x = x, →15 : x → y. → y = y → x. → x hold in it for all x and y. In this case either side of the latter one presents the join of x and y: x ∨ y = x → y. → y.

Moreover, then any pair of elements of A bounded from below has also the meet: for any z ≤ x, y, xy = x → .y → z: → z, and every principal order ﬁlter of A turns out to be a Boolean lattice w.r.t. these operations. See [1, 2] for more information on implication algebras. In [20, 26], Hilbert algebras satisfying →15 were said to be commutative, and those satisfying →13 have also been called Tarski algebras in the literature. Another proper subclass of the class of all Hilbert algebras is that of relatively pseudocomplemented posets [36], i.e., unital posets in which the operation →, following [27], is deﬁned by a → b := max{x: z ≤ a, z ≤ x imply z ≤ b for all z}.

Such posets were called implicative in [33], and Brouwer ordered, in [19]. Relatively pseudocomplemented lower semilattices are known also as implicative semilattices [13, 34], or Brouwerian semilattices. A unital poset (A, ≤, 1) can be turned into a Hilbert algebra also by setting ⎧ ⎪ ⎨ 1ifa ≤ b, a → b := ⎪ ⎩ b otherwise.

This well-known example of a Hilbert algebra goes back to [14, 15]; see also [30]. Hilbert algebras of this kind were recently studied in [7] under the name order algebra; they have also be called pure Hilbert algebras (see, e.g., [3] and reference [3] therein). J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 267

Let (A, →, 1) be some Hilbert algebra. A subalgebra B of A is said to be a block if it is a bounded implication algebra. In this case, if p is the least element of B,then

x = x → p. → p for all x ∈ B. (2)

Theorem 2.1. AsubsetB of A is a block if and only if there is a subalgebra X of A and an element p ∈ X such that the subset X→p := {x → p: x ∈ X} of A coincides with B.

Proof. Every block B coincides with B→p,wherep is the least element of B. Clearly, B→p ⊂ B; on the other hand, B ⊂ B→p in virtue of (2). Now suppose that X is a subalgebra of A and p ∈ X.Thenp is the least element of X→p, and this subset is closed under → : for all x, y ∈ X,

x → p. → .y → p = z → p, where z = x → p. → .y → p: → p ∈ X.

Indeed, x → p. → .y → p ≤ z → p by (1)and→12. The reverse inequality follows by (1) from the identity z → .x → y =1. where x := x → p, y := y → p, z := z → p = x → y. → p: → p :

z → .x → y = x → .z → y (by →9) = x → ::x → y. → p: → p∴ → y = x → .x → y : → .x → p∴ → .x → p:: → .x → y (by →10) = x → y . → .x → p: → .x → p∴ → .x → y (by →8) = x → ∴y → p. → p: → y (by →10) = x → .y → y (by →11)

=1 (by →5, →4) .

Thus, X→p is a bounded subalgebra of A.Moreover,since

x → p. → .y → p: → .x → p ≤ x → p. → p: → .x → p (by →1, →7, →6)

= x → ∴x → p. → p: → p (by →9)

= x → p (by →11, →8), it is an implication algebra — see (→14).

Note that if the subalgebra X is even an implication algebra, then, due to (2), X→p turns out to be a principal order ﬁlter in X: for all x ∈ X, p ≤ x implies that x = y → p with y = x → p . It is common knowledge that the (→, 1)-reduct of an implicative semilattice is a Hilbert algebra. The converse, which is included in the proposition below, is less well- known. 268 J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279

Proposition 2.2. An algebra (A, →, 1) is a Hilbert algebra if and only if it is a subreduct of an implicative semilattice.

Proof. Sufficiency of the condition follows from the above observation. Its necessity is implicit in Theorem 8.5 of [23] (the I-algebras studied there are just the Hilbert algebras). Moreover, as noted by a referee, the proposition is a direct consequence of [14,Theorem 12]. However, we shall discuss some details of the proof from [23], which will be needed below. Let A∗ stand for the set of all finite non-empty subsets of a Hilbert algebra A,every element a of A being identified with its singleton {a};thenA ⊂ A∗ and (A∗, ∪, 1) is an idempotent and commutative monoid freely generated by A;itisconsideredasalower semilattice with unit. Now, the operation → can be extended to A∗ so that

K ∪ L. → M = K → .L → M, K → .L ∪ M = K → L. ∪ .K → M.

It is shown in [23] that such an extension always exists and is deﬁned uniquely (see items 6.1– 6.6 therein). The relation ≤ on A∗ deﬁned by

K ≤ L if and only if K → L =1

is a preorder; moreover, the relation eq deﬁned by

K eq L if and only if K ≤ L and L ≤ K

is even a congruence of the algebra (A∗, →, ∪, 1), and a eq b implies a = b. Therefore, A ⊂ A∗/eq, and the Hilbert algebra A turns out to be the subreduct of the quotient algebra A∗/eq. Finally, it follows from Theorem 7.1 of [23] that the quotient itself is an implicative semilattice.

This proposition and its proof have several useful consequences concerning the lattice of implicative filters of a Hilbert algebra. An implicative filter [35] (or just a filter, or a deductive system) of a Hilbert algebra A is a nonempty subset F such that b ∈ F whenever a → b ∈ F for some a ∈ F .For instance, any principal order filter [a):={x: a ≤ x} is a filter in this sense. In particular, every implicative filter of A contains 1; sometimes this condition is used in the definition instead of being nonempty. Using the notation introduced in the above proof, the filter [X) generated by a subset X of A is described as follows:

[X)={x: K ≤ x for some K ∈ A∗}

—see[14, 15]or,say,[30, Theorem 6], [35, II.3.5]. Note that

[K) ⊂ [L) if and only if L ≤ K. (3)

As usual, ﬁlters of a Hilbert algebra A make up a complete lattice F(A) w.r.t. set inclusion. The lattice is implicative (i.e., a Heyting algebra; see [14, page 42] or [4,Remark J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 269

2.7]) and algebraic [6, Theorem 2]; its compact elements are just finitely generated filters. We shall need the following well-known property of filters, where stands for the join operation in F(A): x → y ∈ F if and only if y ∈ F [x); (4)

see, e.g., [14,Lemma3],[30, Corollary 7] or [4, Corollary 2.5]. Duals of implicative semilattices are called subtractive, or diﬀerence, semilattices [13].

Theorem 2.3. The upper semilattice Ffin(A) of the compact elements of F(A) is a subtractive semilattice, and the subset of principal ﬁlters of A is closed under subtraction.

Proof. Since [K) [L)=[K ∪L), the mapping that sends a subset K into the ﬁlter [K)is ∗ an epimorphism of the semilatice (A , ∪)onto(Ffin(A), ). In virtue of (3), the relation eq is its kernel equivalence; so, the mapping induces an anti-isomorphism f of the lower ∗ semilattice A /eq onto Ffin(A). This anti-isomorphisms turns the latter algebra into a subtractive semilattice; thus, [K) − [L)=[L → K). In particular, [a) − [b)=[b → a), and the set of principal ﬁlters is closed under subtraction.

We note without proof that this theorem is, in fact, equivalent to the assertion that (4) holds for all ﬁnitely generated ﬁlters.

Corollary 2.4. A lattice is isomorphic to the lattice of ﬁlters of a Hilbert algebra if and only if it is isomorphic to the lattice of ﬁlters of an implicative semilattice.

Proof. Suﬃciency of the condition is trivial, as every implicative semilattice is a Hilbert algebra. Its necessity follows from Theorem 3.1 of [34], where a complete lattice L which has a compact base that is a subtractive subsemilattice of L is shown to be isomorphic to the lattice of ﬁlters of the dual of the base.

3 Compatible meets in a Hilbert algebra

Assume that A is a Hilbert algebra, and let ab stand for the meet of elements a and b of A, when it exists. If this is indeed the case, then the following conditions on a and b are equivalent (see [30, Theorem 3]): C1: a → ab = a → b, C2: a ≤ b → ab, C3: ab → x = a → .b → x for all x, C4: for all x,(x → a)(x → b) exists and is equal to x → ab, C5: for all x, ab ≤ x ⇔ a ≤ b → x. Following [30], the elements a and b are said to be compatible (in symbols, aCb), if their meet exists and satisﬁes the ﬁrst, hence, any of the listed conditions. For example, comparable elements are always compatible. It follows from C2 that in a pure Hilbert algebra only comparable elements are compatible. Thus, a Hilbert algebra is pure if and 270 J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279

only if compatibility in it coincides with comparability. Compatibility can be characterized without explicitly referring to meets:

aCbif and only if min{x: a ≤ b → x} exists, (5)

and then the minimum is the meet of a and b.Equivalently,

aCbif and only if there is z such that z ≤ a, z ≤ b and a ≤ b → z;(6)

again, such z is uniquely determined and serves as the meet. Indeed, if aCb,thenC5 shows that ab is the minimum required in (5). Furthermore, if z is the minimum, then z ≤ a, z ≤ b (see →1,resp.,→5) and, for every x, a ≤ b → x implies z ≤ x;thus,the condition at the right hand side of (6) is fulﬁlled. Finally, if z fulﬁlls the condition, then it is even the greatest lower bound of a and b:by→9, →8 and (1), a ≤ b → x ⇒ z ≤ b → x ⇒ b ≤ z → x ⇒ z ≤ z → x ⇒ z ≤ x.

Now aCbby C2. Due to →1 and →11, the relationship

(x → y) C (x → y. → y), (7)

which holds for all x and y, is a self-evident illustration to (6), with y in the role of z. Note that x → y. → y is the complement of x → y in the block A→y;so,(7) is covered by the following more general lemma.

Lemma 3.1. Any two elements of a block are compatible, and their meet in A coincides with their meet in the block.

Proof. Let B be a block in A,andletp be its least element. Suppose that a, b ∈ B and that c is the meet of a and b in B, i.e. c = a → .b → p: → p.Ofcourse,c is a lower bound of a and b also in A. Furthermore, by →9,

a → .b → c = a → ∴a → .b → p: → .b → p = a → .b → p: → :a → .b → p =1,

wherefrom a ≤ b → c by (1). Now (6) applies; so aCb,andc is the meet of a and b.

Theorem 8 of [30] contains the following characteristic of compatible elements in a Hilbert algebra in terms of filters: aCbif and only if [a) [b) is a principal filter. More exactly, [a) [b)=[c) if and only if aCband ab = c. (8) Note that this characteristic explicitly refers only to the order structure of the algebra under consideration. There are good reasons for considering only meets of compatible elements of a Hilbert algebra as “genuine” meets. We define the partial operation ∧ (called compatible meet) on a Hilbert algebra as follows:

a ∧ b := min{x: a ≤ b → x}. (9) J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 271

By (5), a ∧ b exists if and only if aCb. Notice that in a relatively pseudocomplemented Hilbert algebra A all existing meets are compatible. Indeed, if ab exists in A,thenb → ab is the maximum element in {x: z ≤ b, z ≤ x imply z ≤ ab for all z}. Clearly, a belongs to this set, and then aCbby C2. On the other hand, the pure three-element Hilbert algebra with two atoms is not relatively pseudocomplemented in spite of the fact that it has only compatible meets (see above). In [22], Hilbert algebras in which the operation ∧ is total were termed (H)-Hilbert algebras. According to Theorem 5(i) of [30], such an algebra is even an implicative semilattice (in the dual form this was rediscovered in Sect. 3 of [29]); moreover, every implicative semilattice arises in this way. Of course, ab may exist even when a∧b does not. By the way, Hilbert algebras having the meet of every pair of elements have been called Hertz algebras by some authors. In [5, Remark 1.3] an equational axiom system for Hertz algebras is credited to A. Monteiro and H. Porta; these axioms were quoted in [25, 29],andusedin[29] to prove that a Hertz algebra is a (H)-Hilbert algebra. The system of axioms goes back to Monteiro, indeed; however, it was intended to describe the class of implicative semilattices (i.e. (H)-Hilbert algebras!) — see [32]or[13, Exercise 4.C.8]. Therefore, the mentioned result of [29] (Lemma 1 therein) is formally wrong; cf. [10]. For the same reason, Lemma 1.6 and Theorem 2.9 of [25], as well as most results in [5] concerning Hertz algebras are not quite correct: they should be ascribed to implicative semilattices. Actually, use of the term ‘Hertz algebra’ in the mentioned sense is a misunderstanding —see[17] for its history (the term is, in fact, just another name for implicative semilattices) and for respective improvements to [5, 25, 29]. Hilbert semilattices, i.e. Hilbert algebras in which all the meets ab exist, have recently been studied in [16]. By the way, several results of [30, 31] are rediscovered there. Lemma 3.3 below illustrates the role of compatible meets in a Hilbert algebra. A semilattice filter of a Hilbert algebra A is defined to be a nonempty upwards closed subset F that is closed also under existing meets of compatible elements. It follows from C2 that every implicative filter of A is a semilattice filter. Conversely, suppose that F is a semilattice filter and that x ∈ F , x → y ∈ F for some x, y ∈ A.Thenalso x → y. → y ∈ F (see →12). It is easily seen that y is the compatible meet of the elements x → y and x → y. → y (cf. (7)); hence, y ∈ F .Thus,F is also an implicative filter, and we have proved

Lemma 3.2. A subset of a Hilbert algebra is an implicative ﬁlter if and only if it is a semilattice ﬁlter.

It is worthwhile to note that, in virtue of (6), every homomorphism h between Hilbert algebras preserves compatible meets in the following sense:

if aCb, then ha C hb and h(a ∧ b)=ha ∧ hb. (10)

Lemma 3.3. If a Hilbert algebra A is a subreduct of an implicative semilattice B,then, 272 J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279

for all x, y, z ∈ A, z is the compatible meet of x and y in A if and only if it is the meet of x and y in B.

Proof. Suppose that x, y ∈ A.Ifz is the meet of x and y in B,thenz ≤ x, z ≤ y and x ≤ y → z. Therefore, if z ∈ A,thenz is the compatible meet of x and y in A by (6). If, conversely, z ≤ x, z ≤ y and x ≤ y → z for some z in A, then, in B, z ≤ xy ≤ z (for z is the meet of x and y in the subset A). So, z is the greatest lower bound of x and y not only in A, but also in B.

Any Hilbert algebra may be treated as a partial semilattice w.r.t. the operation ∧. Let us give a precise meaning to this observation. First, we adapt to semilattices a term used in [18] for lattices.

Deﬁnition 3.4. A structure (A, ∧)where∧ is a partial binary operation on A, is called a (lower) weak partial semilattice,orawp-semilattice, for short, if it satisﬁes the axioms ∧1 : x ∧ x exists and x ∧ x = x, ∧2 :ifx ∧ y exists, then y ∧ x exists and x ∧ y = y ∧ x, ∧3 :ifx ∧ y. ∧ z and y ∧ z exist, then x ∧ .y ∧ z exists and x ∧ y. ∧ z = x ∧ .y ∧ z. An element 1 is said to be a unit in A if, for all x ∈ A, ∧4 : x ∧ 1 exists and equals to x.

The other, “symmetric” to ∧3, associative law can be obtained as a consequence: if x ∧ .y ∧ z and x ∧ y exist, then y ∧ x exists and x ∧ .y ∧ z = y ∧ z. ∧ x = z ∧ y. ∧ x = z ∧ .y ∧ x = y ∧ x. ∧ z = x ∧ y. ∧ z. Suppose that A is a wp-semilattice. Let aCbnow mean that a ∧ b exists; then the relation ≤ defined by a ≤ b if and only if aCband a ∧ b = a is a partial order on A (with 1 the greatest element), and the element a ∧ b, whenever it exists, is the greatest lower bound (i.e., meet) of a and b ([18, Lemma I.5.16]). We again shall refer to the meets of this kind as compatible. A dual ideal, or a filter,in A is a non-empty upper-subset F of A closed under existing compatible meets. Any principal order filter [a):={x: a ≤ x} is also a filter in this sense. As usual, if A has the unit element, then the filters of A form a complete lattice F(A) with respect to the inclusion ⊂; however, we shall deal only with the upper semilattice (Ffin(A), )ofthose filters generated by a finite subset of A: standard considerations show that Ffin(A)isthe compact basis of F(A). The wp-semilattice A is said to satisfy the condition D (see [18, p. 42]) if [a) [b)=[c) implies aCb; of course, a ∧ b = c in this case. In fact, the following strengthening of the implication holds (cf. (8)): [a) [b)=[c) if and only if aCband a ∧ b = c. (11) J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 273

Proposition 3.5. If a wp-semilattice A with unit satisﬁes the condition D, then the mapping a → [a) is an embedding of A into the dual semilattice of Ffin(A).

The next deﬁnition again is adapted from [18].

Deﬁnition 3.6. A partial semilattice is a relative subalgebra of some meet semilattice L, i.e. a subset A of L with the meet operation restricted to A: for all x, y ∈ A,anelement z of A is the compatible meet of x and y in A if and only if it is the meet of x and y in L. We shall also speak of A as a relative subsemilattice of L in this case.

Every partial semilattice is weakly partial. A full characteristic of partial semilattices follows from the above proposition (cf. [18, Theorem I.5.20]).

Proposition 3.7. Partial semilattices are just those wp-semilattices which satisfy the condition D.

Note that the partial semilattices of [8] are a more specialized notion: they coincide with the wp-semilattices (in fact, posets) having the lower bound property (or, equivalently, the property that every principal ﬁlter is a total meet semilattice). The duals of such algebras were called (upper) nearsemilattices in [9]. We adopt this term here, and call a lower nearsemilattice any wp-semilattice having the lower bound property. For example, every implication algebra is a lower nearsemilattice (see Subsect. 2.1). Notice in connection with the condition D that the lower bound property in a wp-semilattice means the following: [a) [b) ⊂ [c) implies aCb. Now we have achieved the aim set at the beginning of this subsection.

Theorem 3.8. The operation (9) on a Hilbert algebra turns the latter into a partial semilattice with the same induced ordering, ﬁlters, and compatibility relation.

Proof. Assume that A is a Hilbert algebra, and let ∧ be the operation on A defined by (9). According to Proposition 2.2, A is a subreduct of an implicative semilattice. Then (A, ∧) is a partial semilattice by Lemma 3.3. Clearly, the ordering induced by ∧,coincides with the initial ordering of A.By(5), elements a and b are compatible in the partial semilattice if and only if they are compatible in the initial Hilbert algebra. At last, by Lemma 3.2, a subset of A is a filter of the partial semilattice if and only if it is a filter of the Hilbert algebra.

Let us call the partial semilattice arising in a Hilbert algebra A this way the ∧-reduct of A.

Theorem 3.9. A partial semilattice (A, ∧) is the ∧-reduct of a Hilbert algebra (A, →, 1) if and only if both algebras have the same ﬁlters. 274 J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279

Proof. The condition is necessary in virtue of Theorem 3.8. Its suﬃciency follows from (8)and(11).

Consequently, partial semilattices that are ∧-reducts of a Hilbert algebra can be characterized in terms of their ﬁlters: see Theorem 2.3.

4 Implicative partial semilattices

We are now in position to show that a Hilbert algebra is, in fact, a kind of an implicative partial semilattice. The deﬁnition below is patterned after Deﬁnition 3.6.

Deﬁnition 4.1. An implicative partial semilattice, or ip-semilattice, for short, is a relative subsemilattice of an implicative semilattice that is closed under implication.

In more detail, an ip-semilattice is an algebra (A, ∧, →, 1) satisfying the following conditions: (1) (A, ∧, 1) is a partial semilattice with unit, (2) → is a total binary operation on A,and (3) there is an implicative semilattice (B,∧, → 1) such that (A, ∧) is a relative subsemilattice (B,∧), and (A, →, 1) is a subalgebra of (B,→, 1).

If it is the case, we call the operation → an implication on the partial semilattice A. Applying this notion, Proposition 2.2 can be improved as follows.

Theorem 4.2. Every Hilbert algebra, considered as an algebra (A, →, ∧, 1) with ∧ the compatible meet operation, is an implicative partial semilattice, and conversely.

Proof. Suppose that the Hilbert algebra (A, →, 1) is a subreduct of an implicative semilattice B. By Lemma 3.3,the∧-subreduct of A is also a relative subsemilattice of B, and we have arrived at the ﬁrst assertion of the theorem. Conversely, every relative subsemilattice A of an implicative semilattice B, if closed under implication, determines a (→, 1)-subreduct of B that is a Hilbert algebra, while the relativised meet operation of B becomes, according to Lemma 3.3, the compatible meet of A.

The implication in an ip-semilattice can also be characterised directly — by explicit axioms.

Theorem 4.3. Suppose that (A, ∧, 1) is a partial semilattice with unit. A binary operation → on A is an implication if and only if it satisﬁes the conditions i1: if xCy,thenx ≤ y → z ⇔ x ∧ y ≤ z, i2: if x ≤ y → z,theny ≤ x → z, i3: if z ≤ x, z ≤ y and x ≤ y → z,thenxCy, J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 275

i4: if xCythen, for all u, (u → x) C (u → y) and u → x. ∧ .u → y = u → .x ∧ y .

Proof. If A is an ip-semilattice, then, by the previous theorem, we may consider it as a Hilbert algebra. By C5, →9,(1)and(6), it is evident that i1-i3 hold in A.Toprovei4,also C2 and C4 are needed: if xCy,thenx ≤ y → xy,wherefromu → x ≤ u → y. → .u → xy by →7 and →2 and, furthermore, u → xCu→ y. Now assume that the operation → satisfies i1–i4; let us first see, why then (A, →, 1) is a Hilbert algebra. Clearly, xC1; so 1 ≤ y → z if and only if y ≤ z (see i1), as needed for (1). Also, →1 holds in A: due to (1), y ≤ 1=x → x,andthen→1 follows by i2.It remains to prove →2. At first, we shall obtain →6 and →7.Byi2,

y ≤ y → z. → z. (12)

Then x ≤ y implies that x ≤ y → z. → z,and→6 follows by i2. Furthermore, in virtue of →1 y ≤ x → y and y ≤ x → y. → y;nowi3 and i1 yield the inequality x → y. → y: ∧ .x → y ≤ y. Hence, by i1,(12)andi2,

y ≤ z ⇒ x → y. → y: ∧ .x → y ≤ z ⇒ x → y. → y ≤ x → y. → z ⇒ x ≤ x → y. → z ⇒ x → y ≤ x → z,

andwehaveobtained→7,too. Now we proceed as follows: (a) x → y ≤ x → :y → z. → z [(12),→7] (b) x → :y → z. → z∴ → .x → z ≤ x → y. → .x → z [(a),→6] (c) z ≤ y → z. → z [→1] (d) z ≤ y → z [→1] (e) y → zCy→ z. → z [(c),(d),i3] (f) y → z. ∧ :y → z. → z ≤ z [(e),i1] (g) y → z. ∧ :y → z. → z = z [(c),(d),(f)] (h) x → .y → zCx→ :y → z. → z [(e),i4] (i) x → .y → z : ∧ ∴x → :y → z. → z = x → z [(g),(h),i4] (j) x → .y → z ≤ x → :y → z. → z∴ → .x → z [(h),(i),i1] (k) x → .y → z ≤ x → y. → .x → z ; [(j),(b)] therefore, →2 holds. Thus, A is a Hilbert algebra, indeed. Recall that then it has the same compatibility relation as the underlying partial semilattice (Lemma 3.2). To complete the proof, we, in view of Theorem 4.2, have to check that the operation ∧ obeys (9). It follows (by i1) from the assumption xCythat x ≤ y → x ∧ y and that x ≤ y → z implies x ∧ y ≤ z; therefore,

if xCy, then x ∧ y =min{z : x ≤ y → z}. 276 J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279

Conversely, if z0 is the least element in {z : x ≤ y → z},thenz0 ≤ x (by →1), z0 ≤ y (by →5 and →3)andx ≤ y → z0. Henceforth, xCyby i3;moreover,x ∧ y = z0 —see also i1.Thus,(9)holds.

In Sect. 3, lower nearsemilattices, a particular kind of wp-semilattices, were mentioned. In such a wp-semilattice, xCyif and only if x and y have a common lower bound.

Deﬁnition 4.4. A lower nearsemilattice with unit and a binary operation → satisfying i1 and i2 is said to be implicative (or an in-semilattice, for short).

The implicative nearsemilattices, which are just the duals of subtractive nearsemilattices studied in [9], can be characterized also as implicative partial semilattices having the lower bound property. Indeed, it is easily seen that both the condition i3 and the ﬁrst half of i4 are immediate consequences of the lower bound property in an in-semilattice (note that i2 provides the inequalities x ≤ u → x and y ≤ u → y). According to Theorem 14 of [9], the (→, 1)-reduct of an in-semilattice is a Hilbert algebra. Hence, i4 actually holds in any in-semilattice in full by virtue of C4. Our aim in the remainder of the section is to characterize implication operations on partial semilattices alternatively in terms of ﬁlters.

Theorem 4.5. Suppose that (A, ∧, 1) is a partial semilattice with unit and that → is a binary operation on A.Then→ is an implication on A if and only if it satisfies the condition i5: [y → z) ⊂ F ⇔ [z) ⊂ F [y) for every finitely generated filter F of A.

Proof. Necessity of the condition follows from Theorem 4.2 and (4). To prove its suﬃ- ciency, the following particular case of i5 will be useful: i6: [y → z) ⊂ [x) if and only if [z) ⊂ [x) [y). It can also be rewritten as x ≤ y → z if and only if z ∈ [x) [y). (13) If xCy,then[x) [y)=[x ∧ y)by(11); so i1 is included in (13). Also i2 is an immediate consequence of (13). To prove i3,notethatz ≤ x and z ≤ y imply [x) [y) ⊂ [z), while x ≤ y → z implies [z) ⊂ [x) [y) (again, by (13)). Due to the condition (D), xCyas needed. To prove i4, we ﬁrst observe that, for any a, b ∈ A, [a) [a → b)=[a) [b). (14) Indeed, [a) ⊂ [a) [b) and, by i6,[a → b) ⊂ [b) ⊂ [a) [b). Conversely, [a) ⊂ [a) [a → b), and [b) ⊂ [a → b) [a)=[a) [a → b) again by i6.NowletxCy. It follows that [x) [y)=[z), where z stands for x ∧ y.Then [z) ⊂ [x) [y) [u)=[u → x) [u → y) [u) J. C¯ırulis / Central European Journal of Mathematics 5(2) 2007 264–279 277

by (14) and, furthermore, [u → z) ⊂ [u → x) [u → y)byi5. On the other hand, [x) ⊂ [z) ⊂ [z) [u), wherefrom [u → x) ⊂ [u → z)byi6; likewise, [u → y) ⊂ [u → z). Therefore, [u → x) [u → y)=[u → z), and i4 now follows from Condition D.

It follows from the proof that i5 is equivalent to the conjunction of (13)andi4. Moreover, it is sufficient to consider i5 only for filters F generated by sets with not more than two elements. Furthermore, implication on a partial semilattice is uniquely defined: owing to (13),

a → b =max{x: b ∈ [x) [a)} =max{x:[b) ⊂ [x) [a)}.

In particular, this observation settles the similar uniqueness problem for subtraction in upper nearsemilattices posed in [9]. In virtue of Theorems 3.8 and 4.2, the above uniqueness result can be given another form.

Theorem 4.6. Every Hilbert algebra is completely determined by its ∧-reduct.

In other words, this means that the structure of a Hilbert algebra can be speciﬁed in terms of its ordering and compatibility relations alone.

Acknowledgment

This work was supported by Latvian Council of Science, Grant Nr. 05.1881. The author is grateful to the anonymous referees, whose remarks helped to improve the presentation.

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