  The purpose of this paper is to begin studying the computation of the nonzero
determinant of semigroups within the class of finite semigroups that possesses
a pair of non-commutative idempotents. This paper focuses on a class of these
semigroups introduced as $\ll$-smooth semigroups. This computation is
applicable in the context of the extension of the MacWilliams theorem for codes
over semigroup algebras.
