We define algebras of triple semigroup and DeMorgan triple semigroup and by defining three Mann’s compositions and one unary operation on the set of 3-place (ternary) functions over some set, we construct a DeMorgan triple semigroup of 3-place (ternary) functions and so find an abstract characterization of this algebras.

In the present paper we give a partially negative answer to a conjecture of Ghahramani, Runde and Willis. We also discuss the derivation problem for both foundation semigroup algebras and Clifford semigroup algebras. In particular, we prove that if S is a topological Clifford semigroup for which Es is finite, then H1(M(S),M(S))={0}.

In the present paper, we consider biflatness of certain classes of semigroup algebras. Indeed, we give a necessary condition for a band semigroup algebra to be biflat and show that this condition is not sufficient. Also, for a certain class of inverse semi-groups S; we show that the biflatness of L1 (S)” is equivalent to the biprojectivity of L1 (S).

We present a characterization of Arens regular semi-group algebras L1 (S), for a large class of semigroups. Mainly, we show that if the set of idempotents of an inverse semigroupS isfinite, then L1 (S) is Arens regular if and only if S is finite.