scholarly journals Growth sequence of globally idemportent semigroups

1990 ◽  
Vol 48 (1) ◽  
pp. 87-88 ◽  
Author(s):  
György Pollák
Keyword(s):  
Finite Semigroup ◽  
Identity Element ◽  
Growth Sequence

AbstractThe n–th member of the growth sequence of a globally idempotent finite semigroup without identity element is at least 2n. (This had been conjectured by J. Wiegold.)

scholarly journals Growth sequences of finite semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 16-20 ◽  
Author(s):  
James Wiegold ◽  
H. Lausch
Keyword(s):  
Real Number ◽  
Finite Semigroup ◽  
Piecewise Linear ◽  
The Other ◽  
Direct Power ◽  
Growth Sequence ◽  
Group Of Units ◽  
Number Of Generators ◽  
Finite Semigroups ◽  
Minimum Number

AbstractThe growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

scholarly journals ON THE NUMBER OF SUBSEMIGROUPS OF DIRECT PRODUCTS INVOLVING THE FREE MONOGENIC SEMIGROUP

2019 ◽  
Vol 109 (1) ◽  
pp. 24-35
Author(s):  
ASHLEY CLAYTON ◽  
NIK RUŠKUC
Keyword(s):  
Direct Product ◽  
Finite Semigroup ◽  
Identity Element ◽  
Direct Products ◽  
Monogenic Semigroup ◽  
Subdirect Products

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element.

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

Preservation of elementary equivalence under scalar extension

1982 ◽  
Vol 47 (4) ◽  
pp. 734-738
Author(s):  
Bruce I. Rose
Keyword(s):  
Identity Element ◽  
Positive Answer ◽  
General Question ◽  
Division Algebras ◽  
Extension Field ◽  
Elementary Equivalence ◽  
Applied Model ◽  
Finite Dimensional ◽  
Positive Statements ◽  
Finite Dimensional Algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

COMPLEXITY PSEUDOVARIETIES ARE NOT LOCAL: TYPE II SUBSEMIGROUPS CAN FALL ARBITRARILY IN COMPLEXITY

2006 ◽  
Vol 16 (04) ◽  
pp. 739-748 ◽  
Author(s):  
JOHN RHODES ◽  
BENJAMIN STEINBERG
Keyword(s):  
Finite Semigroup ◽  
Type Ii ◽  
Local Type

We prove the following two results announced by Rhodes: the Type II subsemigroup of a finite semigroup can fall arbitrarily in complexity; the complexity pseudovarieties Cn (n ≥ 1) are not local.

scholarly journals A proof of Zhil'tsov's theorem on decidability of equational theory of epigroups

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Inna Mikhaylova
Keyword(s):  
Finite Semigroup ◽  
Equational Theory ◽  
Unary Operation ◽  
Finitely Based ◽  
Infinite Basis ◽  
International Audience

International audience Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.

scholarly journals Finite groups whose intersection power graphs are toroidal and projective-planar

2021 ◽  
Vol 19 (1) ◽  
pp. 850-862
Author(s):  
Huani Li ◽  
Xuanlong Ma ◽  
Ruiqin Fu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Identity Element ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

scholarly journals Some Topological and Polynomial Indices (Hosoya and Schultz) for the Intersection Graph of the Subgroup of〖 Z〗_(r^n )

2021 ◽  
Vol 34 (4) ◽  
pp. 68-77
Author(s):  
Alaa J. Nawaf ◽  
Akram S. Mohammad
Keyword(s):  
Identity Element ◽  
Intersection Graph

         Let  be any group with identity element (e) . A subgroup intersection graph of  a subset  is the Graph with V ( ) =  - e and two separate peaks c and d contiguous for c and d if and only if      , Where  is a Periodic subset of resulting from  . We find some topological indicators in this paper and Multi-border (Hosoya and Schultz) of   , where    ,  is aprime number.

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