Abstract We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant 0-J*$0{\\rm{ - }}{\\cal J}*$-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the ℛ*${\\cal R}*$-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.

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Locally adequate semigroup algebras

Let S be an ℛ-unipotent semigroup such that the idempotent set E(S) is locally ℒ-finite, and let R be a commutative ring with identity. In this paper, we construct a set of orthogonal idempotents {...

ℛ-unipotent semigroup algebras

A construction is given for a trace function on the semigroup algebra of a certain type of E -unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500022654

Trace functions on the algebras of certain E -unitary inverse semigroups

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R0[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.

Semiprimitivity of Orthodox Semigroup Algebras

In 1976, Domanov showed that the algebra of an inverse semigroup S over a field F is semiprimitive (that is, has zero Jacobson radical) if the algebra of each maximal subgroup of S over F is semiprimitive. It is known that the converse statement is false in general. The principal purpose of this paper is to announce that if the semi-lattice of S satisfies a certain finiteness condition, introduced by Teply, Turman and Quesada in 1980, then the converse does hold. Corresponding results for primitivity are also discussed.

A Class of Inverse Semigroup Algebras

In the contracted semigroup algebra A of a finite inverse semigroup S over an arbitrary field there is given a basis, whose elements together with the zero of the algebra A form a semigroup ¯S, which is the 0-direct union of Brandt semigroups, whose description is given in terms of the semigroup S. With the help of this basis there is found the number of nonzero irreducible modular representations of the semigroup S by matrices over an algebraically closed field of nonzero characteristic.

A. V. Rukolaine

Papers

Semigroup algebras of finite inverse semigroups over arbitrary fields