Potential Divisibility in Finite Semigroups is Undecidable

Abstract

We prove that there is no algorithm to decide, given a finite semigroup S and two elements a, b∈S, whether there exists a bigger finite semigroup T>S where a divides b and b divides a. This solves a thirty years old problem by John Rhodes.

The embeddability of ring and semigroup amalgams is undecidable

“…We now state a fundamental lemma concerning split systems constructed in this way. This is proved in [8] and [11]. An analogous result for the related 'split pairs' is implicit in [6].…”

“…If A is a ring (semigroup) amalgam [S\, U; 0,] then we will say A is weakly embeddable in a ring (semigroup) T if for each / there are injective homomorphisms [4] [A]) is (weakly) embeddable in the multiplicative semigroup of R. Thus it will suffice to prove Theorem 1.4 and Theorem 1.5 in the case of semigroups. Theorem 1.4 and Theorem 1.5 admit a relatively simple proof in the style of several other recent undecidability results concerning embedding problems of finite semigroups and general algebras: [3,6,8,9,11]. In particular, the results in [8] can be modified to give Theorem 3.3 in this paper.…”

“…Theorem 1.4 and Theorem 1.5 admit a relatively simple proof in the style of several other recent undecidability results concerning embedding problems of finite semigroups and general algebras: [3,6,8,9,11]. In particular, the results in [8] can be modified to give Theorem 3.3 in this paper. Common to all of these papers are proofs roughly of the following format: for every partial group G a finite semigroup (or algebra) S is constructed such that the answer to the relevant embedding problem is 'yes' for S if and only if G is embeddable in a group.…”

“…As mentioned above, the proof of Theorem 1.4 will use the undecidability of the embedding problem of finite partial groups in the class of all groups and the class of finite groups (see [7,Connection 2.2]). By a partial group we will mean a set with an element 1 and with a partial binary operation which is, where defined, associative and satisfies 1* = x 1 = x and is such that if gh = gk or hg = kg then h = k. A symmetric partial group (see [8]) is a partial group G with the property that for every g € G there is a unique g' e G' such that gg' = g'g = 1. For any finite partial group we may construct a symmetric extension G' of G which is a symmetric partial group containing G such that for every g e G', either g or g' is contained in the set G. This last condition ensures that there are only finitely many possible symmetric extensions and they may be effectively listed.…”

“…https://doi.org/10.1017/S1446788700002214 [8] Theembeddabilityofring 279 LEMMA 2.6. More information on Brandt semigroups and their important role in semigroup theory can be found in [1] or [5].…”