The inverse braid monoid

We introduce an inverse monoid which plays a similar role with respect to the symmetric inverse semigroup that the braid group plays with respect to the symmetric group.     

On the inverse braid monoid

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial n may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations.     

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

Generators and relations for subsemigroups via boundaries in Cayley graphs

Given a finitely generated semigroup S and subsemigroup T of S, we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.     

A finitely presented monoid with a non-finitely generated group of units

We exhibit an example of a monoid defined by finitely many generators and defining relations, the group of units of which is not finitely generated. Received: 30 August 2006 Revised: 23 October 2006     

A note on subsemigroups with divisor theory

The paper is concerned with the problem of describing those subsemigroups of a semigroup with divisor theory which have a divisor theory as well.     

A finite interval in the subsemigroup lattice of the full transformation monoid

In this paper we describe a portion of the subsemigroup lattice of the full transformation semigroup Ω ^Ω , which consists of all mappings on the countable infinite set Ω. Gavrilov showed that there are five maximal subsemigroups of Ω ^Ω containing the symmetric group \(\operatorname {Sym}(\varOmega )\) . The portion of the subsemigroup lattice of Ω ^Ω which we describe is that between the intersection of these five maximal subsemigroups and Ω ^Ω . We prove that there are only 38 subsemigroups in this interval, in contrast to the \(2^{2^{\aleph_{0}}}\) subsemigroups between \(\operatorname {Sym}(\varOmega )\) and Ω ^Ω .     

Groups and actions in transformation semigroups

Let be a transformation semigroup of degree . To each element we associate a permutation group acting on the image of , and we find a natural generating set for this group. It turns out that the -class of is a disjoint union of certain sets, each having size equal to the size of . As a consequence, we show that two -classes containing elements with equal images have the same size, even if they do not belong to the same -class. By a certain duality process we associate to another permutation group on the image of , and prove analogous results for the -class of . Finally we prove that the Schützenberger group of the -class of is isomorphic to the intersection of and . The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper. Received 16 December 1996; in final form 18 February 1997     

Principal topologies and transformation semigroups

For a given set X, the set F(X) of all maps from X to X forms a semigroup under composition. A subsemigroup S of F(X) is said to be saturated if for each x∈X there exists a set Ox⊆X with x∈Ox such that . It is shown that there exists a one-to-one correspondence between principal topologies on X and saturated subsemigroups of F(X). Some properties of principal topologies on X and the corresponding properties of their associated saturated subsemigroups of F(X) are discussed.     

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Let Ω^Ω be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of Ω^Ω, then we write U≈V if there exists a finite subset F of Ω^Ω such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in Ω^Ω is the least cardinality of a subset A of Ω^Ω such that the union of U and A generates Ω^Ω. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω.The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in Ω^Ω where d is the minimum cardinality of a dominating family for N^N. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω^Ω are 0, 1, 2, and d.We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2^ℵ₀.     

Submodules of Kronecker modules via extension monoid products

Using some results on Reineke's extension monoid product over the Kronecker algebra kQ, we provide numerical criteria in terms of Kronecker invariants for a module in $\mathrm{mod}\text{-}kQ$ to be isomorphic with the submodule of an another module in $\mathrm{mod}\text{-}kQ$. The results can be transferred to matrix pencils in solving an important challenge in matrix pencil theory: characterize in terms of Kronecker invariants when a given pencil is a subpencil of an another one.     

Inverse semigroups determined by their lattices of convex inverse subsemigroups I

Every inverse semigroup possesses a natural partial order and therefore convexity with respect to this order is of interest. We study the extent to which an inverse semigroup is determined by its lattice of convex inverse subsemigroups; that is, if the lattices of two inverse semigroups are isomorphic, how are the semigroups related? We solve this problem completely for semilattices and for inverse semigroups in general reduce it to the case where the lattice isomorphism induces an isomorphism between the semilattices of idempotents of the semigroups. For many inverse semigroups, such as the monogenic ones, this case is the only one that can occur. In Part II, a study of the reduced case enables us to prove that many inverse semigroups, such as the free ones, are strictly determined by their lattices of convex inverse subsemigroups, and to show that the answer obtained here for semilattices can be extended to a broad class of inverse semigroups, including all finite, aperiodic ones. Received September 24, 2002; accepted in final form December 15, 2002.     

On a surface singular braid monoid

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follow from topological Yoshikawa moves. As a direct application we will reprove some known theorem about twist-spun knots. We wish then to investigate an index associated to the closure of surface singular braid. Using our relations we will prove that there are exactly six types of knotted surfaces with the index less or equal to two, and there are infinitely many types of surface-knots with index equal to three. Towards the end we will construct a family of classical diagrams such that to unlink them requires at least four Reidemeister III moves.     

奇异保序变换半群的极大正则子半群

设On为通常的有限链Xn={1,2,…,n}上的奇异保序变换半群.文中利用格林关系的方法讨论On的极大正则子半群,确定了On的所有的极大正则子半群.     

Locally maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups

In this paper we study locally maximal idempotent-generated subsemigroups of finite singular orientation-preserving transformation semigroups. This work is supported by N.S.F. and E.S.F. of Guizhou.