Inverse Monoids With a Natural Semilattice Ordering
Jonathan Leech · Proceedings of the London Mathematical Society · 1995
condition is equivalent to S both being (von Neumann) regular and having all idempotents commute. The classic example of such a semigroup is the symmetric inverse semigroup Ix of all partial bijections on a set X under the standard composition of partial functions on X. More generally, for any reasonably endowed mathematical object Q, the set /n of all partial symmetries of Q (bijections between subobjects of Q respecting all relevant structure) forms an inverse semigroup, the symmetric inverse semigroup of Q. Nearly all such semigroups, however, are more than mere inverse semigroups. Clearly such semigroups possess both an identity 1 and a zero 0, so that one has at least inverse monoids with zero; but of greater consequence is the fact that the natural partial order on the semigroup (given by x ^y if and only if y = yy ~ ] x =xy ~ ] y) usually possesses infima of arbitrary non-empty subsets. When this occurs, every element x has a fixed point idempotent f[x] = 1 AX which is maximal among those idempotents lying beneath x in the natural partial order. (For example, in the symmetric inverse semigroup on a set X, infima are given by intersections,