J-trivial Subsemigroups of Finite Full Transformation Semigroups
Tatsuhiko Saito · 1998
Then both of them are subsemigroup of T (X). We call TRE(X,≤) and TOP (X,≤) the full regressive transformation semigroup and the full order-preserving transformation semigroup on X, respectively. It is known (see [3]) that (1) every finite R-trivial semigroup can be embedded in TRE(X,≤) for some totally ordered finite set (X,≤), and (2) every finite J -trivial semigroup divides TRE(X,≤)∩ TOP (X,≤) for some totally ordered finite set (X,≤). When X is a fixed finite set, every R-trivial subsemigroup of T (X) can be embedded in TRE(X,≤) for some total order ≤ on X, and TRE(X,≤) is a unique maximal R-trivial subsemigroup of T (X) up to isomorphisms. However, though TRE(X,≤) ∩ TOP (X,≤) is a J -trivial subsemigroup of T (X), it is not a maximal J -trivial subsemigroup of T (X) even if (X,≤) is a totally ordered set. The purpose of this paper is to determine all maximal J -trivial subsemigroups of T (X) when X is a finite set.