We give lower bounds on the largest singular value of arbitrary matrices, some of which are asymptotically tight for almost all matrices. To study when these bounds are exact, we introduce several combinatorial concepts. In particular, we introduce regular, pseudo-regular, and almost regular matrices. Nonnegative, symmetric, almost regular matrices were studied earlier by Ho¤man, Wolfe, and Ho¤...

We show that the monoid M2(T) of 2 × 2 tropical matrices is a regular semigroup satisfying the semigroup identity A2B4A2A2B2A2B4A2 =A2B4A2B2A2A2B4A2. Studying reduced identities for subsemigroups of M2(T), and introducing a faithful semigroup representation for the bicyclic monoid by 2 × 2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation the...

notions of strongly regular, regular and left(right) regular $gamma$−semigroupsare introduced. equivalent conditions are obtained through fuzzy notion for a$gamma$−semigroup to be either strongly regular or regular or left regular.

We consider a natural generalization of braids which we call shrinking braids. state the relations and use them to define algebraically monoid [Formula: see text]. endow subset text] with left distributive structure it extend Dehornoy order on an By using this order, prove that is isomorphic generated (geometrically) by

The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kau¤man. Borisavljevíc, Došen and Petríc gave a complete proof of its abstract presentation by generators and relations, and suggested the name ‘Kau¤man monoid’. We bring the theory of semigroups to the study of a certain …nite homomorphic image o...

Underlying the notion of an automatic structure is that of a synchronously regular (s-regular for short) set of pairs of strings. Accordingly we consider s-regular preex-rewriting systems showing that even for fairly restricted systems of this form connu-ence is undecidable in general. Then a close correspondence is established between the existence of an automatic structure that yields a preex...