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.

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We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.

We construct a finitely based variety of algebras with two binary operations where the set of subalgebras of finite simple algebras is not recursive. This paper belongs to a series of four papers about unsolvability of algorithmic problems dealing with embeddings of finite semigroups and related algebraic systems. The other papers are [3][4][5]. These papers are based on the result of the first author [3] about undecidability of the set of subsemigroups of ideally simple finite semigroups. Although we use methods from [3], we made this paper self-contained, so it can be read independently of the other papers in the series.It is easy to see that the set of finite simple algebras in any finitely based variety is recursive. In this article, we are interested in the set of subalgebras of finite simple algebras in a variety. The algorithmic problem we are dealing with is the following:Given a finite algebra A in a variety, decide whether A is embeddable into a finite simple algebra in this variety.Every finite group is embeddable into a finite simple group (the group of even permutations on a finite set). Thus our problem is decidable in the variety of all groups. The classification of finite simple groups and the result of Weigel [7] that