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 -smooth semigroups. This computation is applicable in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.
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M. H. Shahzamanian (2024) studied this question.
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