On any set X may be defined the free algebra R〈X〉 (respectively, free commutative algebra R[X]) with coefficients in a ring R. It may also be equivalently described as the algebra of the free monoid X∗ (respectively, free commutative monoid M (X)). Furthermore, the algebra of differential polynomials R{X} with variables inX may be constructed. The main objective of this contribution is to provi...

In this paper we study the variety Mnil of nilpotent elements of a reductive monoid M . In general this variety has a completely different structure than the variety Guni of unipotent elements of the unit group G of M . When M has a unique non-trivial minimal or maximal G×G-orbit, we find a precise description of the irreducible components of Mnil via the combinatorics of the Renner monoid of M...

We show how the categorial approach to inverse monoids can be described as a certain endofunctor (which we call the partialization functor) of some category. In this paper we show that this functor can be used to obtain several recently defined inverse monoids, and use it to define a new object, which we call the inverse braid-permutation monoid. A presentation for this monoid is obtained. Fina...

Let M be a (commutative cancellative) monoid. A nonunit element q ∈ M is called almost primary if for all a, b ∈ M , q|ab implies that there exists k ∈ N such that q|a or q|b. We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties ...

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric...

We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid aer related to the formal language properties of its idempotent problem. In particular,...

In this paper, all the monoids considered are commutative. If S is a monoid generated by {m1, . . . ,mn}, then S is isomorphic to a quotient monoid of N by the kernel congruence σ of the map φ : N → S, φ(k1, . . . , kn) = ∑n i=1 kimi. Under this setting, a finite presentation for S is a finite subset ρ of Nn×Nn such that the congruence generated by ρ is equal to σ. Rédei proves in [5] that ever...

All semigroups and monoids except free monoids and free groups are assumed finite in this paper. A monoid M divides a monoid N if M is a quotient of a submonoid of N . The set of idempotents of a monoid M is denoted E(M). The word “variety” will be used for pseudo-variety — that is, a collection of finite monoids closed under division and finite direct product. A block group is a monoid such th...

In this paper we introduce and study the theories of pseudo links singular in Solid Torus, ST. Pseudo are with some missing crossing information that naturally generalize notion knot diagrams, have potential use molecular biology, while contain a finite number self-intersections. We consider ST set up appropriate topological theory order to construct invariants for these types particular, formu...