Topic
Idempotence
About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.
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TL;DR: In this paper, the authors consider the preservation of properties of being finitely generated, being residually finite under direct products in the context of different types of algebraic structures and identify as broad classes as possible in which the expected preservation results (A × B A × B satisfies property P P if and only if A and B satisfy P P ) may fail outside those classes.
11 citations
Book•
20 Oct 2005
TL;DR: In this article, the reduction to positive characteristic and completion of CG complements are discussed.Motivating Examples: Reduction to Positive Characteristic, Homological Approach, Completions of CG
Abstract: Motivating Examples.- Reduction to Positive Characteristic.- A Homological Approach.- Completions of CG.
11 citations
Posted Content•
TL;DR: This paper gives a single identity of omega-terms for every join level of the Trotter-Weil hierarchy; this yields decidability and shows that the direction alternation depth of unambiguous interval logic is decidable.
Abstract: The variety DA of finite monoids has a huge number of different characterizations, ranging from two-variable first-order logic FO^2 to unambiguous polynomials. In order to study the structure of the subvarieties of DA, Trotter and Weil considered the intersection of varieties of finite monoids with bands, i.e., with idempotent monoids. The varieties of idempotent monoids are very well understood and fully classified. Trotter and Weil showed that for every band variety V there exists a unique maximal variety W inside DA such that the intersection with bands yields the given band variety V. These maximal varieties W define the Trotter-Weil hierarchy. This hierarchy is infinite and it exhausts DA; induced by band varieties, it naturally has a zigzag shape. In their paper, Trotter and Weil have shown that the corners and the intersection levels of this hierarchy are decidable.
In this paper, we give a single identity of omega-terms for every join level of the Trotter-Weil hierarchy; this yields decidability. Moreover, we show that the join levels and the subsequent intersection levels do not coincide. Almeida and Azevedo have shown that the join of R-trivial and L-trivial finite monoids is decidable; this is the first non-trivial join level of the Trotter-Weil hierarchy. We extend this result to the other join levels of the Trotter-Weil hierarchy. At the end of the paper, we give two applications. First, we show that the hierarchy of deterministic and codeterministic products is decidable. And second, we show that the direction alternation depth of unambiguous interval logic is decidable.
11 citations
TL;DR: The concept of a Specker R-algebra is introduced, and it is proved that the Boolean powers of R are up to isomorphism precisely thespecker Ralgebras, and these algebrAs are characterized in terms of a functorial construction having roots in the work of Bergman and Rota.
Abstract: A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker Ralgebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module.
11 citations
TL;DR: In this paper, it was shown that the lattice of pseudovarieties of completely regular semigroup varieties can be decomposed into pseudovarsieties of groups, which are then decompositions of groups.
Abstract: We shall show that several results concerning the lattice of completely regular semigroup varieties find their analogues for the lattice of pseudovarieties of completely regular semigroups. We establish several complete idempotent endomorphisms and a subdirect decomposition of this lattice of pseudovarieties. These investigations culminate in Theorem 18 which is the analogue for pseudovarieties of Polak’s description [19] of the lattice of completely regular semigroup varieties. We shall in particular be able to describe the lattice of pseudovarieties of orthogroups in terms of the lattice of pseudovarieties of groups.
11 citations