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On Distributive Meet-Semilattices

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On Distributive Meet-Semilattices International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 01 | Jan 2020 www.irjet.net p-ISSN: 2395-0072 On Distributive Meet-Semilattices A. Ashtalakshmi1, G. Soundharya2 Thassim Beevi Abdul Kader College for Women, Kilakarai, Ramanathapuram, Tamilnadu, India. -----------------------------------------------------------------***------------------------------------------------------------------ Abstract :- Desire by Gentzen’ s conjunction elimination ruling the Natural Deduction calculus and reading discrimination with join in a natural move, we perceive a perception of Different notions of distributivity for meet-semilattices. In the move we substantiate that those perceptions are linearly ordered. The conjunction -conditional portion of antagonistic logic. Keywords: Meet-semilattice, ND-distributive, K-distributive, H-distributive 1 Introduction Different notions of distributivity for semilattices have been introduced in the article as a ratiocination of the usual distributive property in lattices. As far as we know, notions of distributivity for semilattices have been given. We perceive a notion of distributivity for meet-semilattice. That will be call ND- distributivity. We aspiration to find out whether it is correspondent to any of the perceptions previously present in the article. In doing so, we also analyze the contrasting notion of distributivity for meet-semilattice we have found. Namely, we see that the accustomed perceptions entail each other in the following linear order: GS K (H LR ND) B and we also afford countermodels for the reciprocals. Additionally, we show that H-distributivity may be detect as a very natural adaptation of a move to characterize distributivity for lattice, case that will afford more desire for the use of that perceptions. Indicate that Hickman used the term balmy distributivity for H-distributivity. 2 Preliminaries Definition 2.1 An algebra , is called a lattice if L is a nonempty set, and are binary operation on L. Both and are idempotent, commutative and associative and they satisfy the absorption law. The study of lattices is called the lattice theory. Properties 2.2 An algebra , if L is nonempty set, i. a b = b a ii. a b = b a iii. a (b ) = (a b) c iv. a (b c) = (a b) c v. a (a b) = a vi. a (a b) =a for all a, b, c ϵ L Definition 2.3 A semilattices is a structured S = where . is a binary operations called the semilattices operation such that . is associative (x y) z = x (y z) . is commutative x y = y x . is idempotent xx = x © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2132 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 01 | Jan 2020 www.irjet.net p-ISSN: 2395-0072 Definition 2.4 Distributive join- semilattice are defined dually. A join-semilattice is distributive if for all a, b and x. If X ≤ a b then there exists and such that ≤ a, ≤ b and X= Definition 2.5 A meet-semilattice is distributive if for all a, b and X. If a b ≤ X then there exists and such that a ≤ , b ≤ and X = Definition 2.6 In this section we provide the basic notions and notations that will be used in the paper. Let J = (J, ≤) be a poset. For any J S, we will use the notations and to denote the set lower and upper bound of S, respectively. That is, { } { } Lemma 2.7 Let J = (J, ≤) be a poset. For all a, b, c ϵ J the following statement are equivalent. i. For all x ϵ J, if a ≤ x and b ≤ x, then c ≤ x. ii. { } { } iii. c ϵ { } A poset J = (J, ≤) is a meet-semilattice if inf {a, b} exists for every a, b ϵ J. A poset J = (J, ≤) Is a lattice it is meet-semilattice. As usual the notations a b shall stand for inf {a, b}. Given a meet-semilattice J = (J, ≤), we will use the following notions: J is upwards directed iff for any a, b ϵ J, there exists c ϵ J such that a ≤ c and b ≤ c. A nonempty subset J I is said to be an ideal iff (1) if x, y ϵ I, then x y ϵ I and (2) if x ϵ I and x ≤ y, then y ϵ I. The principal ideal generated by an element a ϵ A, noted (a], is defined by (a] = {x ϵ A: a ≤ x}. Id(J) will denote the set of all ideals of J. (J) will denote the subset of ideals that are union of a finite set of principal ideals, that, is { ] ] } In this paper we are concerned with various notions of distributivity for meet-semilattice, all of them generalizing the usual notion of distributive lattice, that is a lattice J =(J, ≤) is distributive if the following equation holds true for any elements a, b, c ϵ J : (D) a (b c) = (a b) (a c) There are several equivalent formulation of this property, in particular we mention the following ones that are relevant for this paper: For all a, b, c ϵ J, if a b = a c and a b = a c then b = c. © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2133 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 01 | Jan 2020 www.irjet.net p-ISSN: 2395-0072 For any two ideals of J, the ideal generated by their intersection is defined by = {a b : a ϵ , b ϵ }. The set Id(J) of ideals of J is a distributive lattice. For the case of semilattices, several non-equivalent generalizations of these conditions can be found in the article, already mentioned in the introduction. However, as expected, all of them turn to be equivalent to usual distributivity in the case of lattices. In the contrast, in any sense of distributivity for meet-semilattices that coincides with usual distributivity in the case of a lattice, the class of distributive meet-semilattice is not even a quasi-variety. Definition 2.8 A meet-semilattice J = (J, ≤) is called ND-distributive (ND for Natural Deduction) it satisfies ( ). 3 Different notions of distributivity for meet-semilattices 3.1 GS-distributivity The following seems to be the most popular definition of distributivity for meet-semilattices. Definition 1 A meet-semilattice J = (J, ) is GS-distributive iff (GS) for all a, b, x Proposition 1 Let J = (J, ) be a meet-semilattice. Then the following two statements are equivalent: i. Every pair of elements has upper bound. ii. For all a, b, x Proof: (i) Suppose a . Let be a upper bound of {a, x} and be a upper bound of {b, x}. Then a and b . Also, x and x , which implies that x (ii) (i) Let a, b J . We have a b a. Then by hypothesis there exist a , b such that a . As , it follows that a . Then a, b that is is a upper bound of {a, b} 3.2 K-distributivity The concept given in the following definition is similar to the one in (GS). Definition 2 A meet-semilattice J = (J ; ) is K-distributive iff (K) for all a, b, x Proposition 2 GS-distributivity implies K-distributivity, but not conversely. © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2134 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 01 | Jan 2020 www.irjet.net p-ISSN: 2395-0072 Proof The most simple counter-example showing that the reciprocal does not hold is the meet-semilattice in Figure [1], that is not upward directed. Indeed, the given meet-semilattice is K-distributive, as the only way to satisfy the antecedent of (K) is to take a but then the consequent is also true. On the other hand , it is not (GS) distributive, as we have and however , there are no a Figure 1: Meet-semilattice showing that K-does not imply GS-distributivity 3.3 H-distributivity In [10] Hickman introduces the concept of mildly distributive join-semilattices as those join-semilattices whose lattice of their strong ideals is distributive. In [10, Theorem 2.5, p.200] it is stated that it is equivalent in the following statement: (H) for all n and If for all b (if ), Then there exists (x ) The given conditional may be seen as a translation of the following version of distributivity for lattices: If Then In the case of a meet-semilattice J = ( J, ) and using quantifiers, (H) may be rendered as follows: For all n and x, If Then for all y, if for all Then y That is in turn equivalent to: For all n and x, If Then for all y, if (for all z, if x ) Then y Using set-theoretic notation, (H) may also be rendered as follows: (C) for all n and x, If { } { } At this point, the reader may wonder whether the number n of arguments is relevant or whether two arguments are enough. Let us settle this question. Firstly, with that in mind, consider ( ) for all x, If { } { } { } { } { } Now, let us state the following fact. © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2135 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 01 | Jan 2020 www.irjet.net p-ISSN: 2395-0072 Lemma 2 ( ) is equivalent to (C). Proof { } { } .Our goal is to see that y . Then we have { } { } { } , and hence ) suppose { } { } { } . We have to prove that ,if { } { } { } { } hence c Proposition 3 Let J = (J, ) be a meet-semilattice. Then, K-distributivity implies H-distributivity.
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