Computation of the transformation semigroups on three letters

Abstract

Let H denote a set with three elements, and T3 the full transformation semigroup on X, i.e. T3 consists of the twenty-seven self maps of X under functional composition. A transformation semigroup (briefly a τ-semigroup) on three letters is an ordered pair (X, S), where S is any subsemigroup of T3.

Enumerating transformation semigroups

“…The brute-force search space size for T 3 is 2 3 3 = 134 217 728, or approximately 134.2 million. Previous computational investigations further restricted the search to the singular part bringing it down to ≈2.1 million [3,32]. In contrast, using the Minimal Extension method (Section 3.3) together with the Equivalent Generators trick (Section 4.1), only 4344 subsets need to be checked to enumerate the 283 conjugacy classes.…”

“…The basic idea for enumerating by size is to find all valid multiplication tables (up to isomorphism and anti-isomorphism) of the given size [4,5,12,23,24,27,28,30,31]. Our approach here is to enumerate not by size but, rather, by degree of transformation representation [3,32]. Recall that Cayley's Theorem (for semigroups) says that any finite semigroup S is isomorphic to a semigroup of functions on a finite set; the degree of S is defined to be the minimal size of such a set.…”

Constructing Embeddings and Isomorphisms of Finite Abstract Semigroups

“…Similarly, for the question What is computable with n states?, we need a practical algorithm to find isomorphic semigroups. Using the algorithm described here, we extended results of [3,28] and enumerated all 4-state finite computations up to isomorphism [9]. The enumeration was also extended for more general diagram semigroups [11].…”