We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I → M onto the minimal ideal, a congruence on M , and a normal subgroup of a maximal subgroup outside I.Roughly speaking, if α ∈ P n , then α * is obtained by reflecting (a graph representing) α in the horizontal axis midway between the two rows of vertices. It is easy to see that α * * = α, (αβ) * = β * α * , αα * α = α, for all α, β ∈ P n . It follows that P n is a regular * -semigroup, in the sense of Nordahl and Scheiblich [35], with respect to this operation. We have several obvious identities, such as dom(α * ) = codom(α) and ker(α * ) = coker(α). This symmetry/duality will allow us to shorten several proofs. Other diagram monoidsIn this subsection, we introduce a number of important submonoids of the partition monoid P n . Following [33], the Brauer and partial Brauer monoid are defined by B n = {α ∈ P n : all blocks of α have size 2} and PB n = {α ∈ P n : all blocks of α have size at most 2}, Green's equivalences R, L , J , H and D reflect the ideal structure of a semigroup S, and are the fundamental structural tool in semigroup theory. They are defined as follows. We write S 1 = S if S is a monoid; otherwise S 1 is the monoid obtained from S by adjoining an identity element to S. Then, for a, b ∈ S,further, H = R ∩ L , and D is the join R ∨ L : i.e., the least equivalence containing R and L . It is well known that D = R • L = L • R for any semigroup S, and that D = J when S is finite (as is the case for all semigroups considered in this article). If K is any of Green's relations, and if a ∈ S, we write K a = {b ∈ S : a K b} for the K -class of a in S. The set S/J = {J a : a ∈ S} of all J -classes of S is partially ordered as follows. For a, b ∈ S, we say that J a ≤ J b if a ∈ S 1 bS 1 . If T is a subset of S that is a union of J -classes, we write T /J for the set of all J -classes of S contained in T . The reader is referred to [7, Chapter 2], [21, Chapter 2] or [36, Appendix A] for a more detailed introduction to Green's relations.Green's equivalences on all diagram monoids considered in this article are governed by (co)domains, (co)kernels and ranks, as specified in the following proposition. For P n this was first proved in [40], though the terminology there was different. For the other monoids see [10, Theorem 2.4] and also [16][17][18]40]. The proposition will be used frequently throughout the paper without explicit reference.Proposition 2.1. Let K n be any of the monoids P n , PB n , B n , PP n , M n , I n , J n , O n . If α, β ∈ K n , then (i) α R β ⇔ dom(α) = dom(β) and ker(α) = ker(β), (ii) α L β ⇔ codom(α) = codom(β) and coker(α) = coker(β),Remark 2.2. A number of consequences and simplifications ar...

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the nonRees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.

The relative rank rank(S : A) of a subset A of a semigroup S is the minimum cardinality of a set B such that A ∪ B = S. It follows from a result of Sierpiński that, if X is infinite, the relative rank of a subset of the full transformation semigroup T X is either uncountable or at most 2. A similar result holds for the semigroup B X of binary relations on X.A subset S of T N is dominated (by U ) if there exists a countable subset U of T N with the property that for each σ in S there exists µ in U such that iσ iµ for all i in N. It is shown that every dominated subset of T N is of uncountable relative rank. As a consequence, the monoid of all contractions in T N (mappings α with the property that |iα − jα| |i − j| for all i and j) is of uncountable relative rank.It is shown (among other results) that rank(B X : T X ) = 1 and that rank(B X : I X ) = 1 (where I X is the symmetric inverse semigroup on X). By contrast, if S X is the symmetric group, rank(B X : S X ) = 2.

Let P be a partition of a finite set X. We say that a full transformation f : X −→ X preserves (or stabilizes) the partition P if for all P ∈ P there exists Q ∈ P such that P f ⊆ Q. Let T (X, P) denote the semigroup of all full transformations of X that preserve the partition P.In 2005 Huisheng found an upper bound for the minimum size of the generating sets of T (X, P), when P is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture.The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of T (X, P), when P is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then G, a \ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elements g ∈ G generate a semigroup denoted a g | g ∈ G . We classify the finite permutation groups G on a finite set X such that the semigroups G, a , G, a \G, and a g | g ∈ G are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups G, a \ G and a g | g ∈ G are generated by their idempotents for all non-invertible transformations of X.

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To every directed graph E one can associate a graph inverse semigroup G(E), where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C * -algebras, and Toeplitz C * -algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E) \ {0} must be discrete for any directed graph E. On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G(E) admits a T 1 semigroup topology in which G(E) \ {0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.

In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa.Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.