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Polymorphism-homogeneity and universal algebraic geometry

Endre Tóth, Tamás Waldhauser

TL;DR

It is shown that polymorphism-homogeneity is also equivalent to the property that algebraic sets are exactly those sets of tuples that are closed under the centralizer clone of the algebra.

Abstract

We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.

Polymorphism-homogeneity and universal algebraic geometry

TL;DR

It is shown that polymorphism-homogeneity is also equivalent to the property that algebraic sets are exactly those sets of tuples that are closed under the centralizer clone of the algebra.

Abstract

We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.

Paper Structure

This paper contains 13 sections, 22 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Clones and relational clones
  4. Galois connections between operations and relations
  5. Universal algebraic geometry and centralizers
  6. Polymorphism-homogeneity
  7. Injectivity
  8. Polymorphism-homogeneity, algebraic sets and injectivity
  9. Examples
  10. Semilattices
  11. Lattices
  12. Abelian groups
  13. Monounary algebras

Key Result

Theorem 2.1

For any set of operations $F \subseteq \mathcal{O}_A$ and any set of relations $R \subseteq \mathcal{R}_A$, we have $\mathop{\mathrm{Clo}}\nolimits(F) = \mathop{\mathrm{Pol}}\nolimits \mathop{\mathrm{Inv}}\nolimits F$ and $\langle R \rangle_\exists = \mathop{\mathrm{Inv}}\nolimits \mathop{\mathrm{Po

Figures (1)

  • Figure 1: Relationships between property (SDC) and several variants of polymorphism-homogeneity and injectivity.

Theorems & Definitions (40)

  • Theorem 2.1: BKKR1969Geiger1968Romov1981
  • Theorem 2.2: TothWaldhauser2020
  • Proposition 2.3: FarkasovaJakubikova2015PechPech2015
  • Proposition 2.4: PechPech2015
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 30 more