scholarly journals THE SEMIDIRECTLY CLOSED PSEUDOVARIETY GENERATED BY APERIODIC BRANDT SEMIGROUPS

2001 ◽  
Vol 11 (02) ◽  
pp. 247-267 ◽  
Author(s):  
M. LURDES TEIXEIRA
Keyword(s):  
Universal Algebra ◽  
Brandt Semigroup ◽  
Membership Problem ◽  
Finite Semigroups ◽  
Brandt Semigroups

This paper presents a study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B2, denoted V*(B2). We construct a basis of pseudoidentities for the semidirect powers of the pseudovariety generated by B2 which leads to the main result, which states that V*(B2) is decidable. Independently, using some suggestions given by J. Almeida in his book "Finite Semigroups and Universal Algebra", we constructed an algorithm to solve the membership problem in V* (B2).

INTERPRETING GRAPH COLORABILITY IN FINITE SEMIGROUPS

2006 ◽  
Vol 16 (01) ◽  
pp. 119-140 ◽  
Author(s):  
MARCEL JACKSON ◽  
RALPH McKENZIE
Keyword(s):  
Membership Problem ◽  
Finite Semigroups ◽  
Variety Of Semigroups ◽  
Membership Problems

We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colorability problem.

scholarly journals Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups

2016 ◽  
Vol 09 (01) ◽  
pp. 1650021
Author(s):  
Jitender Kumar ◽  
K. V. Krishna
Keyword(s):  
Finite Semigroup ◽  
Brandt Semigroup ◽  
Near Semiring ◽  
Brandt Semigroups

The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of [Formula: see text], the affine near-semiring over a Brandt semigroup [Formula: see text]. It is ascertained that both the semigroup reducts of [Formula: see text] are syntactic semigroups.

scholarly journals IDENTITIES IN BRANDT SEMIGROUPS, REVISITED

2019 ◽  
Vol 5 (2) ◽  
pp. 80
Author(s):  
Mikhail V. Volkov
Keyword(s):  
Brandt Semigroup ◽  
Original Proof ◽  
Main Claim ◽  
Brandt Semigroups ◽  
Finite Exponent ◽  
Made In

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.

POINTLIKE SETS, HYPERDECIDABILITY AND THE IDENTITY PROBLEM FOR FINITE SEMIGROUPS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 475-481 ◽  
Author(s):  
JOHN RHODE ◽  
BENJAMIN STEINBERG
Keyword(s):  
Membership Problem ◽  
Identity Problem ◽  
Nilpotent Semigroup ◽  
Finite Semigroups

In this paper, we give a relationship between the identity problem and the problem of deciding whether certain subsets of nilpotent semigroups are pointlike. We then use this to give an example of a pseudovariety which has a decidable membership problem, but for which one cannot decide pointlike sets. Then, by modifying the equations, we show that no graph is fundamentally hyperdecidable by constructing, for each graph, a labeling over a nilpotent semigroup for which we cannot decide inevitability with respect to the pseudovariety defined by these equations.

UNDECIDABILITY, AUTOMATA, AND PSEUDOVARITIES OF FINITE SEMIGROUPS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 455-473 ◽  
Author(s):  
JOHN RHODES
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Membership Problem ◽  
Joint Paper ◽  
Finite Semigroups ◽  
Membership Problems

The author proves for each of the operations # equalling *, °, **, □ or m, there exist pseudovarieties of finite semigroups [Formula: see text] and [Formula: see text] with decidable membership problems, such that [Formula: see text] has an undecidable membership problem. In addition, if [Formula: see text] denotes the pseudovariety of all finite aperiodic semigroups, [Formula: see text] denotes the pseudovariety of all finite groups, and [Formula: see text](E) denotes the pseudovariety of all finite aperiodic semigroups satisfying the finite number of equations E, then it is proved that there exists E such that [Formula: see text](E) has an undecidable membership problem. Note [Formula: see text] equals all semigroups of complexity ≤1. Section 6 is expanded into a joint paper with B. Steinberg, following this paper.

THE SEMIGROUPS B2 AND B0 ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

2013 ◽  
Vol 23 (06) ◽  
pp. 1289-1335 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Finite Basis ◽  
Brandt Semigroup ◽  
Finitely Based ◽  
Natural Structure ◽  
Locally Finite ◽  
Locally Finite Variety ◽  
Restriction Semigroup ◽  
Inverse Operation ◽  
Brandt Semigroups ◽  
Nonfinitely Based

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids". These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.

scholarly journals Morita Equivalence of Brandt Semigroup Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Maysam Maysami Sadr
Keyword(s):  
Banach Algebras ◽  
Morita Equivalence ◽  
Brandt Semigroup ◽  
Semigroup Algebras ◽  
Homology Groups ◽  
Fixed Group ◽  
Morita Theory ◽  
Morita Invariant ◽  
Brandt Semigroups ◽  
Approximate Amenability

We show that Banach semigroup algebras of any two Brandt semigroups over a fixed group are Morita equivalence with respect to the Morita theory of self-induced Banach algebras introduced by Grønbæk. As applications, we show that the bounded Hochschild (co)homology groups of Brandt semigroup algebras over amenable groups are trivial and prove that the notion of approximate amenability is not Morita invariant.

Finite semigroups and universal algebra

1996 ◽  
Vol 52 (1) ◽  
pp. 393-396
Author(s):  
Boris M. Schein
Keyword(s):  
Universal Algebra ◽  
Finite Semigroups
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