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.

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 Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

2020 ◽  
Vol 31 (04) ◽  
pp. 527-538
Author(s):  
Grzegorz Madejski ◽  
Andrzej Szepietowski
Keyword(s):  
Computational Model ◽  
Turing Machine ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Membership Problem ◽  
Two Dimensional ◽  
Np Complete ◽  
Nondeterministic Turing Machine ◽  
Membership Problems

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

Diving into the queue

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 89-110
Author(s):  
Simon Beier ◽  
Martin Kutrib ◽  
Andreas Malcher ◽  
Matthias Wendlandt
Keyword(s):  
Finite Number ◽  
Finite Automata ◽  
Time Constraints ◽  
Membership Problem ◽  
Storage Medium ◽  
Logarithmic Space ◽  
Better Than

We introduce and study the model of diving queue automata which are basically finite automata equipped with a storage medium that is organized as a queue. Additionally, two queue heads are provided at both ends of the queue that can move in a read-only mode inside the queue. In particular, we consider suitable time constraints and the case where only a finite number of turns on the queue is allowed. As one main result we obtain a proper queue head hierarchy, that is, two heads are better than one head, and one head is better than no head. Moreover, it is shown that the model with one queue head, finitely many turns, and no time constraints as well as the model with two queue heads, possibly infinitely many turns, and time constraints is captured by P and has a P-complete membership problem. We obtain also that a subclass of the model with two queue heads is already captured by logarithmic space. Finally, we consider decidability questions and it turns out that almost nothing is decidable for the model with two queue heads, whereas we obtain that at least emptiness and finiteness are decidable for subclasses of the model with one queue head.

FC-nilpotent and FC-soluble groups

1956 ◽  
Vol 52 (3) ◽  
pp. 391-398 ◽  
Author(s):  
A. M. Duguid ◽  
D. H. McLain
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Abelian Groups ◽  
Characteristic Subgroup ◽  
Arbitrary Group ◽  
Soluble Groups ◽  
Image Position

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G byHi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

scholarly journals Translation planes of odd order and odd dimension

1979 ◽  
Vol 2 (2) ◽  
pp. 187-208 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Translation Planes ◽  
Solvable Subgroup ◽  
Joint Paper ◽  
Desarguesian Planes ◽  
Odd Order ◽  
Collineation Groups ◽  
Zero Vector

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension2d overGF(q), whereqanddare odd. If the stabilizer of the zero vector is non-solvable, letG0be a minimal normal non-solvable subgroup. We suspect thatG0must be isomorphic to someSL(2,u)or homomorphic toA6orA7. Our main result is that this is the case whendis the product of distinct primes.The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow2-groups whendandqare both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order27(i.e.,dandqare both equal to3) which admitsSL(2,13).

Coset Enumeration in a Finitely Presented Semigroup

1978 ◽  
Vol 21 (1) ◽  
pp. 37-46 ◽  
Author(s):  
Andrzej Jura
Keyword(s):  
Finite Number ◽  
Finite Index ◽  
Finite Groups ◽  
Formal Proof ◽  
Finitely Presented Group ◽  
Group B ◽  
Coset Enumeration ◽  
Finitely Presented

The enumeration method for finite groups, the so-called Todd-Coxeter process, has been described in [2], [3]. Leech [4] and Trotter [5] carried out the process of coset enumeration for groups on a computer. However Mendelsohn [1] was the first to present a formal proof of the fact that this process ends after a finite number of steps and that it actually enumerates cosets in a group. Dietze and Schaps [7] used Todd-Coxeter′s method to find all subgroups of a given finite index in a finitely presented group. B. H. Neumann [8] modified Todd-Coxeter′s method to enumerate cosets in a semigroup, giving however no proofs of the effectiveness of this method nor that it actually enumerates cosets in a semigroup.

COMPUTATIONALLY AND ALGEBRAICALLY COMPLEX FINITE ALGEBRA MEMBERSHIP PROBLEMS

2007 ◽  
Vol 17 (08) ◽  
pp. 1635-1666 ◽  
Author(s):  
MARCIN KOZIK
Keyword(s):  
Finite Algebra ◽  
Membership Problem ◽  
Membership Problems

In this paper we produce a finite algebra which generates a variety with a PSPACE-complete membership problem. We produce another finite algebra with a γ function that grows exponentially. The results are obtained via a modification of a construction of the algebra A(T) that was introduced by McKenzie in 1996.

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.

scholarly journals On a bottom layer in a group

2020 ◽  
Vol 100 (4) ◽  
pp. 136-142
Author(s):  
V.I. Senashov ◽  
◽  
I.A. Paraschuk ◽  
◽  
Keyword(s):  
Finite Elements ◽  
Finite Number ◽  
Finite Index ◽  
Finite Groups ◽  
Prime Order ◽  
Bottom Layer ◽  
Arbitrary Group ◽  
Locally Finite

We consider the problem of recognizing a group by its bottom layer. This problem is solved in the class of layer-finite groups. A group is layer-finite if it has a finite number of elements of every order. This concept was first introduced by S. N. Chernikov. It appeared in connection with the study of infinite locally finite p-groups in the case when the center of the group has a finite index. S. N. Chernikov described the structure of an arbitrary group in which there are only finite elements of each order and introduced the concept of layer-finite groups in 1948. Bottom layer of the group G is a set of its elements of prime order. If have information about the bottom layer of a group we can receive results about its recognizability by bottom layer. The paper presents the examples of groups that are recognizable, almost recognizable and unrecognizable by its bottom layer under additional conditions.

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