A ring (R, *) with involution * is called formally complex if implies that all Ai are 0. Let (R, *) be a formally complex ring and let S be an inverse semigroup. Let (R[S], *) be the semigroup ring with involution * defined by . We show that (R[S], *) is a formally complex ring. Let (S, *) be a semigroup with proper involution *(aa* = ab* = bb* ⇒ a = b) and let (R, *′) be a formally complex ring. We give a sufficient condition for (R[S], *′) to be a formally complex ring and this condition is weaker than * being the inverse involution on S. We illustrate this by an example.

A semigroup 5 with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S, (3t G T)(Vu, v 6 T) tt* = uv' => u = v. It is shown that a semigroup is inverse if and only if it is a special involution semigroup in which every element invariant under the involution is periodic. Other examples of special involution semigroups are discussed; these include free semigroups, totally ordered cancellative commutative semigroups and certain semigroups of matrices. Some properties of the semigroup algebras of special involution semigroups are also derived. In particular, it is shown that their real and complex semigroup algebras are semiprimitive. 1. DEFINITIONS

background

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Formal complexity of inverse semigroup rings

On semigroups with involution

In this paper, it is shown that a cancellative semigroup is embeddable in an inverse semigroup. It is shown that finite proper *-semigroup is regular and any finite commutative proper *-semigroup is a union of groups. Also it is shown that a finite cyclic proper * semigroup is a group while an infinite one is *-embedded in a proper*-group, and any finite maximal proper*- semigroup has a proper *-extension ring. It is shown that there is a nonregular proper *-ring that cannot be *-embedded in any regular proper *-ring. Also it is shown that an Artinian proper *-ring is a finite direct product of matrix rings over skew fields. It is shown that a commutative proper * and cancellative semigroup is *-embeddable in a regular proper *-semigroup.

Regular proper *-embedding of proper *-semigroups and rings

Let S be an inverse semigroup. We prove that there is a ring with a proper involution * in which S is *-embeddable. The ring will be a natural one, R[S], the semigroup ring of S over any formally complex ring R; for example ℝ, Ȼ.

Formal complexity of inverse semigroup rings

Citations

On semigroups with involution

Regular proper -embedding of proper -semigroups and rings

References

Proper embeddability of inverse semigroups

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