Recursive axiomatizations for representable posets

Abstract

We define a fragment of monadic infinitary second-order logic corresponding to a kind of abstract separation property. We use this to define certain subclasses of elementary classes as separation subclasses. We use model theoretic techniques and games to show that separation subclasses which are, in a sense, recursively enumerable in our second-order fragment can also be recursively axiomatized in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications, we use simple characterizations as separation subclasses to obtain axiomatizability results related to graph colourings and partial algebras.2010 Mathematics Subject Classification. Primary 03C98. Secondary 03B15, 03B70, 05C15, 08A55.

Recursive Axiomatisations From Separation Properties

“…Such a non-constructive proof of existence may be regarded as being of limited practical use, however, the very fact that an axiomatisation is known to exist can be used in a neat trick to show that a certain constructively generated axiomatisation is correct. This is the main result of [15].…”

“…We say a poset P is representable if there is a set X and an order embedding such that h preserves finite meets and joins from P whenever they exist here is considered as a lattice with operations and . It is easy to prove that a poset P is representable if and only if whenever there is an -filter of P containing p and not see, for example, [15, Theorem 2.4]). Thus, building on Example 2.4, we see that the class of representable posets is an essentially r.e.…”

“…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.…”

“…The method of [15], which is not novel, is to describe the ‘separation property’ of representable posets in terms of a game played between two players. The game is defined so that the number of rounds a certain player can survive in a particular game corresponds, in a sense, to how close a given poset is to being representable.…”

“…The main purpose of this paper is to prove a general theorem that includes the relevant results of [15, 24] as special cases, and is also applicable in a wide variety of other situations. The strategy is to first formalise the concept of a ‘separation property’ in a way that allows the necessary results to go through, while also being intuitive enough to be useful in practice.…”