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Enumeration of idempotents in planar diagram monoids
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- and edge-coloured graphs
associated to Motzkin diagrams. The enumeration is necessarily algorithmic in
nature, and is based on parameters associated to cycle components of these
graphs. We compare our algorithms to existing algorithms for enumerating
idempotents in arbitrary (regular *-) semigroups, and give several tables of
calculated values.Comment: Majorly revised (new title, new abstract, one additional author), 24
pages, 6 figures, 8 tables, 5 algorithm
Variants of finite full transformation semigroups
The variant of a semigroup S with respect to an element a in S, denoted S^a,
is the semigroup with underlying set S and operation * defined by x*y=xay for
x,y in S. In this article, we study variants T_X^a of the full transformation
semigroup T_X on a finite set X. We explore the structure of T_X^a as well as
its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a
(consisting of all products of idempotents), and the ideals of Reg(T_X^a).
Among other results, we calculate the rank and idempotent rank (if applicable)
of each semigroup, and (where possible) the number of (idempotent) generating
sets of the minimal possible size.Comment: 25 pages, 6 figures, 1 table - v2 includes a couple more references -
v3 changes according to referee comments (to appear in IJAC
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