Computing Embeddings and Isomorphisms of Finite Semigroups

TL;DR

This work tackles the problem of determining embeddings and isomorphisms between finite semigroups, which model state-transition computations and relate to the graph-isomorphism problem. It introduces partitioned backtrack, a backtracking approach augmented by index-period invariants and other algebraic properties to dramatically prune search space, enabling new computational results for transformation and diagram semigroups. The authors develop parallelization, conjugation-aware enumeration, generator-based morphism construction, and custom diagram representations, and demonstrate capabilities through minimal-degree realizations, embedding counts, and 2-generated subsemigroup embedding questions. The outcomes advance understanding of how different diagram representations compare in expressive power and provide practical tools and data for semigroup theory, automata theory, and algebraic computing. The work enhances the practical feasibility of embedding and isomorphism tasks in nontrivial semigroups and offers guidance for future algorithmic and software improvements with broad implications for algebraic automata theory.

Abstract

Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering questions like `What is the minimal number of states to realize a particular computation?' and `Which type of computation is more capable?' translate into the algebraic tasks of constructing isomorphisms and embeddings between semigroups of different representations. The underlying problem is (sub)graph isomorphism, which is computationally difficult in general. We describe variations of backtrack search algorithms that exploit the algebraic properties of semigroups, and we carry out computational experiments to extend our algebraic knowledge. In particular, we report new computational results on transformation semigroups and on the more general family of diagram semigroups. We study the minimal degree representation problem, count distinct embeddings and work on an open problem of embedding into 2-generated subsemigroups.

Figures (5)

• Figure 1: (left) Composing partitioned binary relations $\alpha$ and $\beta$. The arrows of the product are induced by paths of the stacked diagram with the property that the consecutive arrows have alternating colors. (right) Embeddings of same degree diagram semigroups.
• Figure 2: Example of a semigroup element with index-period (3,2). The third power $s^3$ is the least power that gets repeated, as $s^5=s^3$. All higher powers alternate between $s^3$ and $s^4$ in a 2-cycle.
• Figure 3: Multiplication in the Brauer monoid $\mathfrak B_4$. In our custom representation $\alpha=[6,5,4,3,2,1,8,7]$, $\beta=[7,8,4,3,6,5,1,2]$ and the product is $\alpha\beta=[8,7,4,3,6,5,2,1]$. 'Loops' formed in the middle are ignored. In general, we may need to trace long paths of alternating black and white edges.
• Figure 4: An example matrix semigroup with listed elements and the corresponding abstract multiplication table. The numbers in the table correspond to the order of the listed matrices.
• Figure 5: All 4 (up to conjugation) embeddings of $\mathcal{T}_3$ into $\mathcal{T}_4$ given by the mappings of the standard generators. The embeddings simply fix the additional point, or collapse it into another point. For higher degree embeddings, avoiding the interference with the multiplication in $\mathcal{T}_3$ yields combinatorial structures of similar fixing and collapsing.

Theorems & Definitions (8)

• Example 2.1
• Definition 2.2: Index-period
• Definition 2.3: $\cong_\multimap$
• Example 2.4
• Definition 2.5: $\cong_\boxplus$
• Example 2.6
• Definition 5.1
• Example 5.2