A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.

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We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister-Schreier type rewriting.

We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.

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...