CREAM: a Package to Compute [Auto, Endo, Iso, Mono, Epi]-morphisms, Congruences, Divisors and More for Algebras of Type $(2^n,1^n)$

TL;DR

CREAM is a versatile software package that computes automorphisms, congruences, endomorphisms, and subalgebras for a wide range of algebraic structures using universal algebra principles, AI theorem proving, and optimized code.

Contribution

It introduces a general-purpose, efficient GAP package capable of handling diverse algebraic structures without relying on domain-specific theorems.

Findings

CREAM outperforms existing algorithms for most algebra classes.

It is the first to compute many objects for numerous algebra types.

The package achieves significant speedups over prior methods in various cases.

Abstract

The CREAM GAP package computes automorphisms, congruences, endomorphisms and subalgebras of algebras with an arbitrary number of binary and unary operations; it also decides if between two such algebras there exists a monomorphism, an epimorphism, an isomorphism or if one is a divisor of the other. Thus it finds those objects for almost all algebras used in practice (groups, quasigroups in their various signatures, semigroups possibly with many unary operations, fields, semi-rings, quandles, logic algebras, etc). As a one-size-fits-all package, it only relies on universal algebra theorems, without taking advantage of specific theorems about, eg, groups or semigroups to reduce the search space. Canon and Holt produced very fast code to compute automorphisms of groups that outperform CREAM on orders larger than 128. Similarly, Mitchell et al. take advantage of deep theorems to compute automorphisms and congruences of completely 0-simple semigroups in a very efficient manner. However these domains (groups of order above 128 and completely 0-simple semigroups) are among the very few examples of GAP code faster than our general purpose package CREAM. For the overwhelming majority of other classes of algebras, either ours is the first code computing the above mentioned objects, or the existing algorithms are outperformed by CREAM, in some cases by several orders of magnitude. To get this performance, CREAM uses a mixture of universal algebra algorithms together with GAP coupled with artificial intelligence theorem proving tools (AITP) and very delicate C implementations. As an example of the latter, we re-implement Freese's very clever algorithm for computing congruences in universal algebras, in a way that outperforms all other known implementations.