Categories of partial algebras for critical points between varieties of algebras

TL;DR

This paper develops a categorical framework for partial algebras called gamps, demonstrating how certain lifting problems in algebraic varieties relate to the existence of semilattices with specific properties, and constructing examples with particular extension properties.

Contribution

It introduces a new categorical theory of gamps for partial algebras and applies it to analyze lifting problems and congruence-preserving extensions in algebraic varieties.

Findings

Existence of semilattices with specific lifting properties in different varieties.

Construction of a lattice with no congruence n-permutable, congruence-preserving extension.

Semilattice S of cardinality aleph d with particular lifting characteristics.

Abstract

A lifting of a semilattice S is an algebra A such that the semilattice of compact (=finitely generated) congruences of A is isomorphic to S. The aim of this work is to give a categorical theory of partial algebras endowed with a partial subalgebra together with a semilattice-valued distance, that we call gamps. This part of the theory is formulated in any variety of (universal) algebras. Let V and W be varieties of algebras (on a finite similarity type). Let P be a finite lattice of order-dimension d>0. Assume that we have a diagram of semilattice with a lifting in V, but with no "partial lifting" in the category of gamps of W, then there is a semilattice S of cardinality aleph (d - 1), such that S has a lifting in V, but S has no lifting in W. We already knew a similar result for diagrams with no lifting in W, however the semilattice S constructed here has cardinality aleph d. Gamps are also used to study congruence-preserving extensions. Denote by M the variety generated by the lattice of length two, with three atoms. We construct a lattice A in M of cardinality aleph 1 with no congruence n-permutable, congruence-preserving extension, for each n > 1.