University of Waikato 2005-09-13 G1.15 On any coset of a group one can define a ternary operation
[xyz]:=xy –1z. In this way one obtains a heap, which
satisfies the following axioms:
- [xxx]=x (idempotence)
- [[vwx]yz]=[v[yxw]z]=[vw[xyz]] (para-associativity)
- [xyy]=x; [xxy]=y
- hence also [[xyy]zz]=[[xzz]yy], [xx[yyz]]=[yy[xxz]].
Weaker structures have been studied: for example, a generalised
heap satisfies (1), (2) and (4) above but not necessarily (3) (which
roughly corresponds to weakening "group" to "inverse semigroup"). A
semiheap satisfies only the para-associativity condition.
Idempotent semiheaps (satisfying just 1 and 2 above) do not seem to
have been considered much.
In the first half of this talk, we give a broad background for the
area, and discuss some of the known representation theorems. We then
consider a class of idempotent semiheaps of which a typical example is
all partial maps X → Y for which [fgh] is f restricted
to where g, h agree, and obtain a representation in terms of such
examples. These algebras can also be viewed as enrichments of
restrictive semigroups.
In the second half, we look at the general theory
of idempotent semiheaps. All idempotent semiheaps can be represented
using suitably chosen pairs (f, f ) of functions between
two sets, where f is a generalised inverse to f
with ternary operation
(f,f)(g,g)(h, h)
= (fgh,
hgf).
We will sketch the outline of a proof of this result using
left and right regular representations of an idempotent semiheap.
We also look at the question of when and how these can arise from
semigroups with additional unary operation, and will completely
characterise the idempotent semiheaps
which arise as subsemiheaps of an I-semigroup with involution. This
latter class turns out to be a variety containing all generalised
heaps.
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