Efficient Enumeration of Higher Order Algebraic Structures

Abstract

Algebraic structures are widely studied mathematical structures in abstract algebra. Enumerating higher order algebraic structures is a computationally intensive task due to large number of possible permutations and the presence of many symmetrically equivalent redundant structures. This paper describes a comprehensive methodology and a novel algorithm to efficiently enumerate higher order algebraic structures. Enumeration of these structures is performed in two steps, namely: generation of complete set of algebraic structures with given constraints, and then computationally demanding task of performing isomorphism checking to identify isomorphism classes. In this paper we choose to study inverse property loops (IP loops), a particular class of algebraic structures, but the methodology can be applied to enumerate any algebraic structure with given constraints. IP loop constraints are modeled in Google's or-tools constraint solver to generate the complete set of IP loops of given order. The paper then discusses and evaluates several techniques to efficiently identify isomorphism classes within these algebraic structures. A novel algorithm is proposed that utilizes valid mappings and a tree data structure for efficient isomorphism checking. The algebraic structures are also modeled as color graphs and isomorphism classes are determined using state of the art graph isomorphism checking tool (nauty). The performance of these proposed isomorphism checking approaches is then evaluated using a diverse set of algebraic structure problems. It also presents an efficient use of multi-core systems to further enhance the efficiency of isomorphism checking. The proposed approach was then used to solve the previously computationally unsolved problem of determining isomorphism classes of exponent 3 IP loops of order 15. INDEX TERMS Algebraic structures, constraint solver, IP loops, isomorphism checking, isomorphism classes, nauty.

SAT-Based Techniques for Lexicographically Smallest Finite Models

“…It is possible to construct a canonical form of a magma by using nauty: for a magma A, construct a special graph G ′ A and find its canonical graph G A , cf. (Khan 2020). This form is canonical in the sense that two isomorphic magmas will give the same canonical graph but the resulting graph is opaque to the user.…”