Automatic Rees Matrix Semigroups over Categories

Abstract

Abstract. We consider the preservation of the properties of automaticity and prefix-automaticity in Rees matrix semigroups over semigroupoids and small categories. Some of our results are new or improve upon existing results in the single-object case of Rees matrix semigroups over semigroups.

Automatic semigroups and categories

“…While the proofs of the lemmas are rather technical, the conditions for application are quite intuitive, and can be easier to check than, for example, the existence of a synchronous (in a suitable sense analogous to that implicitly used in [9]) rational transducer computing a given function. Another application of these results will be seen in the subsequent paper [23], where they will be used to study the relationship between automaticity and Rees matrix constructions over semigroupoids.…”

“…More recently, Lawson [24] has introduced a method of finding abundant semigroups, which employs a form of Rees matrix construction over semigroupoids. In a subsequent article [23] we shall use the theory developed in this paper to show that, under certain assumptions, the property of automaticity passes through these constructions.…”

Uniform decision problems for automatic semigroups

“…Define a new alphabet It can now be shown (for example, using arguments similar to those in [16,Proof of Theorem 4.6]) that each of these relations is synchronously rational and straightforward to compute. This completes the construction of an automatic structure for the maximal subgroup H i 0 λ 0 .…”

“…Let L(∆) = φ(L ∩ R). It follows from[16, Proof of Theorem 4.6] that L(∆) is a regular language and is straightforward to compute. Now recalling that P λ 0 i 0 = e λ 0 i 0 , one can show thatL = (∆) = {(uφ, vφ) | (u, v) ∈ L = ∩ (R × R)} while for each c a ∈ B we have L ca (∆) = {(uφ, vφ) | (u, v) ∈ L wa ∩ (R × R)}and for each d λi ∈ B L d λ i (∆) = {(uφ, vφ) | (u, v) ∈ L P λi ∩ (R × R)}.…”