Permutation identity near-rings and ?localized? distributivity conditions

1991 ◽  
Vol 111 (4) ◽  
pp. 265-285 ◽  
Author(s):  
Gary Birkenmeier ◽  
Henry Heatherly
Keyword(s):  
Permutation Identity

scholarly journals Some saturated varieties of semigroups

1983 ◽  
Vol 27 (3) ◽  
pp. 419-425 ◽  
Author(s):  
N.M. Khan
Keyword(s):  
Sufficient Conditions ◽  
Permutation Identity

We show that a semigroup satisfying a heterotypical identity of which at least one side has no repeated variable is saturated and find sufficient conditions on a homotypical identity which is not a permutation identity and of which at least one side has no repeated variable, to ensure that any semigroup satisfying the identity is saturated.

Commutative orders

1996 ◽  
Vol 126 (6) ◽  
pp. 1201-1216 ◽  
Author(s):  
David Easdown ◽  
Victoria Gould
Keyword(s):  
Homomorphic Image ◽  
Commutative Semigroup ◽  
Permutation Identity ◽  
Left Order ◽  
Right Order

A subsemigroup S of a semigroup Q is a left (right) order in Q if every q ∈ Q can be written as q = a*b(q = ba*) for some a, b ∈S, where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. If S is both a left order and a right order in Q, we say that S is an order in Q. We show that if S is a left order in Q and S satisfies a permutation identity xl…xn = x1π…xnπ where 1 < 1π and nπ<n, then S and Q are commutative. We give a characterisation of commutative orders and decide the question of when one semigroup of quotients of a commutative semigroup is a homomorphic image of another. This enables us to show that certain semigroups have maximum and minimum semigroups of quotients. We give examples to show that this is not true in general.

scholarly journals Minimal ideals in near-rings and localized distributivity conditions

1993 ◽  
Vol 54 (2) ◽  
pp. 156-168 ◽  
Author(s):  
Gary Birkenmeier ◽  
Henry Heatherly
Keyword(s):  
Exact Sequence ◽  
Subdirect Product ◽  
Minimal Ideal ◽  
Basic Properties ◽  
Permutation Identity ◽  
Made In

AbstractLet B, S, and T be subsets of a (left) near-ring R with B and T nonempty. We say B is (S, T)-distributive if s(b1+b2)t = sb1t + sb2t, for each s ∈ S, b1, 2 ∈ B, t ∈ T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals. Theorem. If I is a minimal ideal of R and Ik is (Im, In)-distributive for some k, n ≧ 1, m ≧ 0, then either I2 = 0 or I is a simple, nonnilpotent ring with every element of I distributive in R. Theorem. Let Rk be (Rm, Rn)-distributive, for some k, n ≧ 1, m ≧ 0; if R is semiprime or is a subdirect product of simple near-rings, then R is a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. If R → A → 0 is an exact sequence of near-rings, then conditions on A are given which will impose conditions on the minimal ideals of R.

Triangularization of Matrices and Polynomial Maps

2019 ◽  
Vol 63 (1) ◽  
pp. 94-105
Author(s):  
Yueyue Li ◽  
Yan Tian ◽  
Xiankun Du
Keyword(s):  
Permutation Identity ◽  
Polynomial Maps

AbstractWe present conditions for a set of matrices satisfying a permutation identity to be simultaneously triangularizable. As applications of our results, we generalize Radjavi’s result on triangularization of matrices with permutable trace and results by Yan and Tang on linear triangularization of polynomial maps.

On Uσ-Abundant Semigroups

2012 ◽  
Vol 19 (01) ◽  
pp. 41-52 ◽  
Author(s):  
Xueming Ren ◽  
Qingyan Yin ◽  
K. P. Shum
Keyword(s):  
Structure Theorem ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Permutation Identity ◽  
Spined Product ◽  
Abundant Semigroups ◽  
The Relationship ◽  
Ehresmann Semigroups

A U-abundant semigroup whose subset U satisfies a permutation identity is said to be Uσ-abundant. In this paper, we consider the minimum Ehresmann congruence δ on a Uσ-abundant semigroup and explore the relationship between the category of Uσ-abundant semigroups (S,U) and the category of Ehresmann semigroups (S,U)/δ. We also establish a structure theorem of Uσ-abundant semigroups by using the concept of quasi-spined product of semigroups. This generalizes a result of Yamada for regular semigroups in 1967 and a result of Guo for abundant semigroups in 1997.

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