We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid Kn are nonfinitely based for each n ≥ 3. This result holds also for the case when Kn is considered as an involution semigroup under either of its natural involutions.(M. V. Volkov

Abstract. We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis. Background and motivation

Abstract. We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition.We define a pseudovariety of groups H to be arboreous if all finitely generated free pro-H groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties H, a pro-H analog of the Ribes and Zalesskiȋ product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions H to the much studied pseudovariety equation J * H = J m H.

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As a step in a study of the lattice ℒ(ℱ) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, a family of complete congruences is introduced on ℒ(ℱ). Such congruences provide a framework from which to study ℒ(ℱ) both locally and globally. Each is associated with a mapping of the form [Formula: see text] for some special class [Formula: see text] of finite semigroups. In some instances the class is itself a pseudovariety while in others the class will be defined in terms of certain congruences associated with Green's relations. The basic properties of these complete congruences are presented and some relations between certain operators associated with these congruences are obtained.