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Algebraic Aspects of Relatively Pseudocomplemented Posets

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Algebraic Aspects of Relatively Pseudocomplemented Posets Order (2020) 37:1–29 https://doi.org/10.1007/s11083-019-09488-1 Algebraic Aspects of Relatively Pseudocomplemented Posets Ivan Chajda1 · Helmut Langer¨ 1,2 · Jan Paseka3 Received: 21 November 2017 / Accepted: 12 April 2019 / Published online: 1 May 2019 © The Author(s) 2019 Abstract In Chajda and Langer¨ (Math. Bohem. 143, 89–97, 2018) the concept of relative pseudo- complementation was extended to posets. We introduce the concept of a congruence in a relatively pseudocomplemented poset within the framework of Hilbert algebras and we study under which conditions the quotient structure is a relatively pseudocomplemented poset again. This problem is solved e.g. for finite or linearly ordered posets. We charac- terize relative pseudocomplementation by means of so-called L-identities. We investigate the category of bounded relatively pseudocomplemented posets. Finally, we derive certain quadruples which characterize bounded Hilbert algebras and bounded relatively pseu- docomplemented posets up to isomorphism using Glivenko equivalence and implicative semilattice envelope of Hilbert algebras. Keywords Relative pseudocomplementation · Poset · Hilbert algebra · Congruence · Convex poset · Dedekind-MacNeille completion · Glivenko equivalence · Category Support of the research of all authors by the Austrian Science Fund (FWF), project I 1923-N25, and the Czech Science Foundation (GACR),ˇ project 15-34697L, as well as support of the research of the first two authors by OAD,¨ project CZ 02/2019, and support of the research of the first author by IGA, project PrFˇ 2019 015, is gratefully acknowledged. Jan Paseka [email protected] Ivan Chajda [email protected] Helmut Langer¨ [email protected] 1 Faculty of Science, Department of Algebra and Geometry, Palack´y University Olomouc, 17. listopadu 12, 771 46, Olomouc, Czech Republic 2 Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria 3 Faculty of Science, Department of Mathematics and Statistics, Masaryk University, Kotla´rskˇ a2,´ 611 37, Brno, Czech Republic 2 Order (2020) 37:1–29 1 Introduction The origin of pseudocomplemented and relatively pseudocomplemented lattices and semi- lattices can be found in papers quoted in the references in the monographs by H.B. Curry [17, 18] and the history of this topic goes back to Skolem as noted in [18], where rela- tively pseudocomplemented lattices are investigated under the name “implication lattices”. Some part of the theory is contained in the paper [1] by R. Balbes. As to pseudocom- plemented lattices, they have been investigated formerly by V. Glivenko and M.H. Stone, pseudocomplemented semilattices were introduced and studied by O. Frink [21]. The early work on relative pseudocomplementation in posets was initiated by T. Katrinˇak´ [23, 24]. We cannot omit the paper [33] by S. Rudeanu where he proves that every relatively pseudocomplemented poset is a Hilbert algebra w.r.t. relative pseudocomplementation. The concept of relative pseudocomplementation plays a crucial role in intuitionistic logic where it models logical implication. Due to this fact, relative pseudocomplementation was investigated by several authors, e.g. by P. Kohler¨ [25] and W. C. Nemitz [29] in their pio- neering works. It turns out that relative pseudocomplementation can be extended to posets. A general approach in this direction was already presented by the first two authors in [7, 10]. An alternative approach was presented by the first author in [6]. The aims of the present paper are as follows: – to get conditions on a congruence in a relatively pseudocomplemented poset such that the quotient structure of the corresponding Hilbert algebra induced by a congruence is a relatively pseudocomplemented poset again, – to characterize relative pseudocomplementation by means of so-called L-identities, – although the class of relatively pseudocomplemented posets does not form a variety, it forms a category whose properties are investigated, – to introduce the so-called characterizing quadruple which enables us to reconstruct the given bounded relatively pseudocomplemented poset. At first we recall some basic concepts. Let A = (A, ≤) be a non-empty poset and a,b,c ∈ A. For every subset B of A we define U(B) := {y ∈ A | x ≤ y for all x ∈ B} and L(B) := {x ∈ A | x ≤ y for all y ∈ B}. Then (U, L) is the Galois correspondence induced by ≤ and L(Bi) = L( Bi) for any i∈I i∈I family Bi,i ∈ I, of subsets of A.IfB ={a,b}, we will write briefly L(a, b) or U(a,b)for L({a,b}) or U({a,b}), respectively, and L(a) instead of L(a,a). A subset of a poset A is up-directed if it is non-empty and every pair of elements has an upper bound in the subset. A poset A is an up-directed-complete poset if each of its up-directed subsets has a supremum. If A has a least element 0 then an element b of A is called the pseudocomplement of a if b is the greatest element x of (A, ≤) satisfying L(a, x) ={0}. In this case we denote b by a∗. The poset A is called pseudocomplemented (or a Boolean ordered set) if each element a of A has a pseudocomplement (and a∗∗ = a, respectively). We denote by B(A) the set {x ∈ A | x = x∗∗} of Boolean elements. Our study is based on extensively making use of results reached in Hilbert algebras by several authors. Hence, we repeat the basic definitions and concepts. We will use, instead of the classical operation → the operation ∗ since our main motivation comes from relatively pseudocomplemented posets. Order (2020) 37:1–29 3 A Hilbert algebra A = (A, ∗, 1) (see [19, 20, 32–34]) may be treated as a poset (A, ≤) with the greatest element 1 equipped with a binary operation ∗ such that x ∗ y = 1 if and only if x ≤ y, x ≤ y ∗ x, x ∗ (y ∗ z) ≤ (x ∗ y) ∗ (x ∗ z) . A will be called bounded if it has a smallest element 0; in this case, for x ∈ A we put x∗ := x ∗ 0. The class H (H0) of all (bounded) Hilbert algebras considered as algebras of the form (A, ∗, 1) ((A, ∗, 1, 0), respectively) is equationally definable. Note that a homomorphism of Hilbert algebras is exactly a mapping between Hilbert algebras preserving operations ∗ and 1 (and hence preserving also order). Since any Boolean algebra is a bounded Hilbert algebra we will further use the notation (B, ∗, 1, 0) for Boolean algebras as well. An element c of a poset A = (A, ≤) is called the relative pseudocomplement of a with respect to b, in symbols c = a ∗ b ([10]), if c is the greatest element x of A satisfying L(a, x) ⊆ L(b).Ifx ∗ y exists for all x,y ∈ A then the poset A is called relatively pseudocomplemented and ∗ is said to be a relative pseudocomplementation on A. Relatively pseudocomplemented meet-semilattices are known also as implicative semilattices [18, 29], or Brouwerian semilattices. S. Rudeanu has shown in [33] that the class of all relatively pseudocomplemented posets is a proper subclass of the class of all Hilbert algebras. Hence we may work further in the context of Hilbert algebras. Definition 1.1 Let (A, ∗, 1) be a Hilbert algebra. We say that (A,a ∗, 1) is relatively pseu- docomplemented poset if a ∗ b is the greatest element of the set {x ∈ A | L(a, x) ⊆ L(b)} for all a,b ∈ A. Let R (R0) denote the class of all (bounded) relatively pseudocomplemented posets. A homomorphism of (bounded) relatively pseudocomplemented posets is a homomorphism of (bounded) Hilbert algebras. Obviously, in the case when A is a meet-semilattice we have that L(a, x) ⊆ L(b) is equivalent to a ∧ x ≤ b and hence the concept of relative pseudocomplement in posets is a generalization of the corresponding concept for meet-semilattices introduced in [18]. Definition 1.2 An implicative semilattice is an algebra (A, ∧, ∗, 1) of type (2, 2, 0) such that (A, ∧) is a meet-semilattice and (A, ∗, 1) is a relatively pseudocomplemented poset. Let IS (IS0) denote the class of all (bounded) implicative semilattices. A homomor- phism of (bounded) implicative semilattices is a homomorphism of (bounded) Hilbert algebras that preserves finite meets. We will write UI for the forgetful functor from IS to H.FixA ∈ H,apair(G,e),where G is an implicative semilattice and e is an injective homomorphism from A to UI(G),issaid to be an implicative semilattice envelope of A if for every y ∈ G there exists a finite subset X ⊆ A such that y = e(X). It is well known (see [5]) that there is a functor S from H to IS such that, for any A ∈ H, we have an injective homomorphism e from A to UI(S(A)) and (S(A), e) is an implicative semilattice envelope of A. In what follows we will always assume that A ⊆ S(A), i.e., A will be a Hilbert subalgebra of UI(S(A)) and e an inclusion. 4 Order (2020) 37:1–29 The following theorem summarizes some properties from [5], of interest for the present work: Theorem 1.3 There exists a functor S: H → IS that maps every Hilbert algebra A to an implicative semilattice S(A), and every homomorphism h: A → B of Hilbert algebras to a homomorphism S(h): S(A) → S(B) of implicative semilattices such that S(h)( X) = h(X) for any finite subset X ⊆ A. The functor S is a left adjoint to UI. Let A = (A, ∗, 1) be a Hilbert algebra. The compatibility relation C on Hilbert algebras was introduced in [27]. We will use an equivalent definition from [14]: Elements a,b ∈ A are said to be compatible (in symbols, ab¸) if they have a lower bound c such that a ≤ b ∗ c.
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