PROFINITE IDENTITIES FOR FINITE SEMIGROUPS WHOSE SUBGROUPS BELONG TO A GIVEN PSEUDOVARIETY

2003 ◽  
Vol 02 (02) ◽  
pp. 137-163 ◽  
Author(s):  
J. ALMEIDA ◽  
M. V. VOLKOV
Keyword(s):  
Finite Groups ◽  
Finitely Based ◽  
Finite Semigroups

We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite pro-identity basis for the pseudovariety [Formula: see text] of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of finite groups.

ON THE IDENTITIES OF REPRESENTATIONS OF FINITE GROUPS OVER COMMUTATIVE RINGS

1992 ◽  
Vol 02 (01) ◽  
pp. 103-116
Author(s):  
SAMUEL M. VOVSI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Finite Basis ◽  
Finitely Based ◽  
Commutative Noetherian Ring ◽  
Representations Of Finite Groups ◽  
A Finite Group ◽  
Finite Exponent

Let K be a commutative noetherian ring. It is proved that a representation of a finite group on a K-module of finite length or on a K-module of finite exponent has a finite basis for its identities. In particular, this implies an earlier result of Nguyen Hung Shon and the author stating that every representation of a finite group over a field is finitely based. The problem whether every representation of a finite group over a commutative noetherian ring is finitely based still remains open.

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.

FINITELY BASED, FINITE SETS OF WORDS

2000 ◽  
Vol 10 (06) ◽  
pp. 683-708 ◽  
Author(s):  
MARCEL JACKSON ◽  
OLGA SAPIR
Keyword(s):  
Direct Product ◽  
Free Monoid ◽  
Finitely Based ◽  
Finite Sets ◽  
Finite Semigroups ◽  
Finite Set ◽  
Rees Quotient ◽  
The Ideal

For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This resulting monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U⊃W and V⊃W such that the identities satisfied by S(V) are finitely based and those of S(U) are not finitely based [regardless of the situation for S(W)]. The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example.

scholarly journals On a problem of M. P. Schützenberger

1980 ◽  
Vol 23 (3) ◽  
pp. 243-247 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Finite Groups ◽  
Direct Products ◽  
Finite Semigroups ◽  
The University

A class of finite semigroups is called a genus if it is closed under homomorphic images, subsemigroups and finite direct products. During a talk at the Symposium on Semigroups held at the University of St Andrews, in 1976, M. P. Schützenberger posed the problem of characterising the smallest genus which contains finite groups and finite semigroups, all of whose subgroups are trivial.

scholarly journals Linear Conjugacy

2019 ◽  
Vol 62 (4) ◽  
pp. 886-895
Author(s):  
Benjamin Steinberg
Keyword(s):  
Finite Groups ◽  
Linear Representation ◽  
Arbitrary Field ◽  
Finite Semigroups

AbstractWe say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.

scholarly journals The embeddability of ring and semigroup amalgams is undecidable

2000 ◽  
Vol 69 (2) ◽  
pp. 272-286 ◽  
Author(s):  
Marcel Jackson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Rings ◽  
Marked Contrast ◽  
Finite Semigroups ◽  
A Finite Group

AbstractWe prove that there is no algorithm to determine when an amalgam of finite rings (or semigroups) can be embedded in the class of rings or in the class of finite rings (respectively, in the class of semigroups or in the class of finite semigroups). These results are in marked contrast with the corresponding problems for groups where every amalgam of finite groups can be embedded in a finite group.

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