Representation of Posets

Abstract

In this paper we give new criterions for left distributive posets to have neatest representations. We also illustrate a construction that would embed left distributive posets into representable semilattices. MSC: 06A10, 06A12.Any poset (P, 5) has at least two representations [l] as a poset (Q, S). One of these two representations has the property that the image of the supremum of any (finite or infinite) subset A of P is the union of the images of the elements of A, while the other representation has the property that the image of the infimum of any (finite or infinite) subset B of P is the intersection of the images of the elements of B. In general, we cannot have a representation which has both of the above mentioned properties. However, Smm (71 proved that i f ( P , 5) is a complemented distributive lattice, i.e., a Boolean Algebra, then it has a representation as a set (Q, E). BIRKHOFF (21 has proved a Stone type representation theorem where the lattice (P, 5) is only distributive. Along these lines, SCHEIN [6] proved that a poset in which the iniimum of every two elements exists and which has a special type of distributive property has a representation as a set (Q, E). There are other types of representations given in [3], [4], and[5]. In Section 1 of this paper, we give some other necessary and suficient conditions for posets with this type of distributivity to be representable. In Section 2, we generalize SCHEIN'S result and finally, in Section 3, we illustrate a construction that would embed existing posets into SCHEIN'S type of semilattices by adjoining infimum. With this procedure one could then show the representability of more general posets. 1. Neatest representations of posets D e f i n i t i o n 1. An order isomorphism f from a poset (P, 5) onto a poset (a S ) is called a neatest representation for P iff for every subset {ai: i E S) of P and for every u, b E P (i) /(Aic ai) = nie J ( a i ) , whenever A;. ui exists, (ii) f(o v b) = f ( a ) u f ( b ) , whenever a v b exists D e f i n i t i o n 2 . Let (P, s) be a poset. (P, s) is lefr meet distributive (1.m.d.) iff for every 19 ZUchr. f. math. Logik

Representable posets

“…A step in this argument assumes the existence of finite meets, which is invalid in the more general setting of posets. This is a symptom of a deeper problem, as a simple generalization of the axiom schema for semilattices (such as appears as the condition LMD in [5], see definition 1.4 below) is not expressive enough for the poset situation (see example 1.5, where it is shown that LMD is not a necessary condition for a poset to have a representation). Note that the question of whether LMD is a sufficient condition for representability with respect to all meets and binary joins is raised as open in [5], and appears not to have been resolved.…”

Recursive Axiomatisations From Separation Properties

“…A partially ordered set (poset) is representable if it can be embedded into a powerset algebra via a map that preserves existing finite meets and joins. The class of representable posets and its infinitary variations have been studied, not always using this terminology, in [8, 11–15, 21, 29, 39], generalising work done in the setting of semilattices [2, 9, 27, 33], and for distributive lattices and Boolean algebras [1, 3, 4, 6, 7, 16, 31, 35, 36]. At first glance, it is far from obvious that the class of representable posets is elementary.…”

Recursive axiomatizations for representable posets

“…A partially ordered set (poset) is representable if it can be embedded into a powerset algebra via a map that preserves existing finite meets and joins. The class of representable posets (RP) and its infinitary variations have been studied, not always using this terminology, in [8,28,20,38,11,13,12,14,15], generalizing work done in the setting of semilattices [2,32,9,26], and for distributive lattices and Boolean algebras [3,35,30,34,4,6,7,1,16]. At first glance, it is far from obvious that RP is an elementary class.…”