IDEALS IN LEFT ALMOST SEMIGROUPS

2015

In this work, we attempt to investigate the connection between various types of ideals (for examples (m;n)-ideal, bi-ideal, interior-ideal, quasi-ideal, prime-ideal and maximal-ideal) of an or- dered semigroup (S; � ; � ) and the corresponding, hyperideals of its EL-hyperstructure (S; � ) (if exists). Moreover, we construct the class of EL--semihypergroup, associated to a partially-ordered - semigroup.

Asian-European Journal of Mathematics, 2009

In this paper we define prime, semiprime and irreducible ideals in ternary semigroups. We also define semisimple ternary semigroups and prove that a ternary semigroup is semisimple if and only if each of its ideals is semiprime.

Semigroup Forum, 1991

Semigroup Forum, 2017

A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574-584, 2005). Keywords Right chain ordered semigroup • Prime ideal • Completely prime ideal • Semiprime ideal • Completely semiprime ideal • Prime segment

arXiv (Cornell University), 2022

A semigroup S is said to be right pseudo-finite if the universal right congruence can be generated by a finite set U ⊆ S × S, and there is a bound on the length of derivations for an arbitrary pair (s, t) ∈ S × S as a consequence of those in U . This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green's pre-orders ≤ L or ≤ J is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or J -triviality.

Semigroup Forum, 1981

Dedicated to L. r~'. GLUSI<IN 3n the occasion of his 60th birthday.

In this paper the terms pseudo symmetric ideals, pseudo symmetric ternary semigroups, semipseudo symmetric ideals and semipseudo symmetric ternary semigroups. It is proved that for any pseudo symmetric ideal A in a ternary semigroup T, for any natural number n, a 1 a 2 … a n-1 a n ∈ A if and only if < a 1 > < a 2 > ………… < a n > ⊆ A. It is proved that every completely semiprime ideal of a ternary semigroup is a pseudo symmetric ideal. Further it is proved that an ideal A of a ternary semigroup is (1) completely prime iff A is prime and pseudo symmetric, (2) completely semiprime iff A is semiprime and pseudo symmetric. It is also proved that every prime ideal P minimal relative to containing a pseudo symmetric ideal A in a ternary semigroup T is completely prime and hence every prime ideal P minimal relative to containing a completely semiprime ideal A in a ternary semigroup T is completely prime. It is proved that every pseudo commutative ternary semigroup, ternary semigroup in which every element is a mid unit, are pseudo symmetric ternary semigroups. It is proved that every pseudo symmetric ideal of a ternary semigroup is a semipseudo symmetric ideal. It is also proved that every semiprime ideal P minimal relative to containing a semipseudo symmetric ideal A of a ternary semigroup is completely semiprime. If A is a semipseudo symmetric ideal of a ternary semigroup T, then A 1 = the intersection of all completely prime ideals of T containing A, (2) 1 A = the intersection of all minimal completely prime ideals of T containing A, (3) 1 A = the minimal completely semiprime ideal of T relative to containing A, (4) A 2 = {x ∈ T : x n ∈ A for some odd natural number n},(5) A 3 = the intersection of all prime ideals of T containing A, (6) 3 A = the intersection of all minimal prime ideals of T containing A, (7) 3

Journal of Algebra, 2005

Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Multiplicative semigroups of right chain rings, right cones, right invariant right holoids and right valuation semigroups are examples. The ideal theory of right chain semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given, extending to these semigroups results on right cones proved by Brungs and Törner [H.H. Brungs, G. Törner, Ideal theory of right cones and associated rings, J. Algebra 210 (1998) 145-164].

2015

We prove that if S is a le-semigroup in which left ideal elements commute (condition which is called Λ), then any J-class satisfying the Green condition is a subsemigroup of S. As a corollary of this we show that semisimple le-semigroups satisfying Λ are precisely those that decompose as a semilattice of left simple ∨ e-semigroups which are in addition semisimple, intra-regular and satisfy Λ.