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Complete representation by partial functions for signatures containing antidomain restriction

Brett McLean

TL;DR

For 14 expressively distinct signatures, it is shown, by giving an explicit representation, that the (meet-)completely representable algebras form a basic elementary class, axiomatisable by a universal-existential-universal first-order sentence.

Abstract

We investigate notions of complete representation by partial functions, where the operations in the signature include antidomain restriction and may include composition, intersection, update, preferential union, domain, antidomain, and set difference. When the signature includes both antidomain restriction and intersection, the join-complete and the meet-complete representations coincide. Otherwise, for the signatures we consider, meet-complete is strictly stronger than join-complete. A necessary condition to be meet-completely representable is that the atoms are separating. For the signatures we consider, this condition is sufficient if and only if composition is not in the signature. For each of the signatures we consider, the class of (meet-)completely representable algebras is not axiomatisable by any existential-universal-existential first-order theory. For 14 expressively distinct signatures, we show, by giving an explicit representation, that the (meet-)completely representable algebras form a basic elementary class, axiomatisable by a universal-existential-universal first-order sentence. The signatures we axiomatise are those containing antidomain restriction and any of intersection, update, and preferential union and also those containing antidomain restriction, composition, and intersection and any of update, preferential union, domain, and antidomain.

Complete representation by partial functions for signatures containing antidomain restriction

TL;DR

For 14 expressively distinct signatures, it is shown, by giving an explicit representation, that the (meet-)completely representable algebras form a basic elementary class, axiomatisable by a universal-existential-universal first-order sentence.

Abstract

We investigate notions of complete representation by partial functions, where the operations in the signature include antidomain restriction and may include composition, intersection, update, preferential union, domain, antidomain, and set difference. When the signature includes both antidomain restriction and intersection, the join-complete and the meet-complete representations coincide. Otherwise, for the signatures we consider, meet-complete is strictly stronger than join-complete. A necessary condition to be meet-completely representable is that the atoms are separating. For the signatures we consider, this condition is sufficient if and only if composition is not in the signature. For each of the signatures we consider, the class of (meet-)completely representable algebras is not axiomatisable by any existential-universal-existential first-order theory. For 14 expressively distinct signatures, we show, by giving an explicit representation, that the (meet-)completely representable algebras form a basic elementary class, axiomatisable by a universal-existential-universal first-order sentence. The signatures we axiomatise are those containing antidomain restriction and any of intersection, update, and preferential union and also those containing antidomain restriction, composition, and intersection and any of update, preferential union, domain, and antidomain.
Paper Structure (8 sections, 21 theorems, 32 equations, 3 figures, 2 tables)

This paper contains 8 sections, 21 theorems, 32 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Algebras of partial functions and their representations
  3. Complete representations
  4. Atomicity
  5. Distributivity
  6. A representation
  7. Axiomatising the classes
  8. Open problems

Key Result

Theorem 2.4

Let $\{\mathbin \vartriangleright'\} \subseteq \sigma \subseteq \{\mathbin \vartriangleright', \wedge, [\space x], \sqcup\}$ or $\{\mathbin \vartriangleright', \mathbin{;}\} \subseteq \sigma \subseteq \{\mathbin \vartriangleright', \mathbin{;}, \wedge, [\space x], \sqcup, \mathop{\mathrm{D}}\nolimit

Figures (3)

  • Figure 1: Hasse diagram of the signatures investigated and their relative expressiveness. Each vertex represents the signature containing the operations appearing below the vertex.
  • Figure 2: An algebra refuting right-distributivity over meets
  • Figure 3: Algebra for which $\theta$ does not represent range correctly

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Jackson and Stokes 10.1093jigpaljzac058
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • ...and 44 more