International Journal of Algebra and Computation

Let n ≥ 2 be a natural number, X = { x₁,…, x_n} and let F be the free group, freely generated by X. Let R be a cyclically reduced word in F such that its symmetric closure in F satisfies the small cancellation condition C′(1/5) & T(4). Let G be the group presented by . A Magnus subsemigroup of G is any subsemigroup of G generated by at most 2n - 1 elements of . In this paper we solve the Membership Problem for rational subsets of G which are contained in a Magnus subsemigroup of G, provided that satisfies certain combinatorial conditions. We use small cancellation theory with word combinatorics.

We study a variety of modes (idempotent algebras with mutually commuting term operations), so called differential modes, having a strongly solvable chain 0 ≤ α ≤ 1 in their congruence lattices. We show an explicit description of subdirectly irreducible algebras in this variety, and use it to compute residual bounds of its subvarieties. It follows from our results that all subvarieties with a finite residual bound are finitely based.

Let G be a finite group and α be an automorphism of G of order pⁿ for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p′-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = {a₁,…, a_r} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {a_l+1,…, a_r} for some l < r, then S is a subset of a basis for the relatively free group on {a₁,…, a_l}.

We describe the two-generated limits of abelian-by-(infinite cyclic) groups in the space of marked groups using number theoretic methods. We also discuss universal equivalence of these limits.

A group G is said to have finite section p-rankr_p(G) = r (here p is a prime) if every elementary abelian p-section U/V of G is finite of order at most p^r and there is an elementary abelian p-section A/B of G such that |A/B| = p^r. If ℙ is the set of all primes and λ : ℙ → ℕ ∪ {0} is a function, we say that a group G has λ-bounded section rank if r_p(G) ≤ λ(p) for each p ∈ ℙ. In this paper we show that if G is a locally generalized radical group in which the normal closures of the cyclic subgroups of G have finite λ-bounded section rank, then [G,G] has -bounded section rank for some function . This is a wide generalization of some results by Neumann, Smith and many others. Moreover we are able to give explicit formulas for the involved bounds.

In this paper we study zero-divisor graphs of rings and semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle.

This paper further develops the theory of arbitrary semigroups acting on trees via elliptic mappings. A key tool is the Lyndon–Chiswell length function L for the semigroup S which allows one to construct a tree T and an action of S on T via elliptic maps. Improving on previous results, the length function of the action will also be L.

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contains, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson.

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here that it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed.

It is shown that a graph product of right-orderable groups is right orderable, and that a graph product of (two-sided) orderable groups is orderable. The latter result makes use of a new way of ordering free products of groups.

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classification of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a difficult problem in general. A restricted problem, also of significant interest, is to ask this question for unary automatic presentations: automatic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally finite, but non-finitely generated unary FA-presentable semigroups may be infinite. Every unary FA-presentable semigroup satisfies some Burnside identity. We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classification is given of the unary FA-presentable completely simple semigroups.