We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the number of idemptents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions.Expand

In this paper we survey various classifications of symmetric games and their characterisations under the theme of fairness; show that game bijections and game isomorphisms form groupoids; introduce matchings as a convenient characterisation of strategy triviality; and outline how to construct and partially order parameterised asymmetric games with numerous examples that range over various classes.Expand

We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex-… Expand

We describe a refinement of the natural partial order on the set of idempotents of the Motzkin monoid of general degree n which exhibits its structure as a disjoint union of geometric complexes. This… Expand

In this paper we survey various notions of symmetry for finite strategic-form games; show that game bijections and game isomorphisms form groupoids; introduce matchings as a convenient… Expand

In this paper we survey various notions of symmetry for finite strategic-form games; show that game bijections and game isomorphisms form groupoids; introduce matchings as a convenient… Expand

In this paper we survey the various notions of anonymity and symmetry for finite strategic-form games that are present in the literature, and discuss notions of fairness; show that game bijections… Expand

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set $X\subseteq\mathbb Z^2$, there are two naturally associated… Expand