Right Product Quasigroups and Loops (original) (raw)
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Right Product Quasigroups and Loops - Journal version
2014
Right groups are direct products of right zero semigroups and groups and they play a signicant role in the semilattice decomposition theory of semigroups. Right groups can be characterized as associative right quasigroups (magmas in which left translations are bijective). If we do not assume associativity we get right quasigroups which are not necessarily representable as direct products of right zero semigroups and quasigroups. To obtain such a representation, we need stronger assumptions which lead us to the notion of right product quasigroup. If the quasigroup component is a (one-sided) loop, then we have a right product (left, right) loop. We nd a system of identities which axiomatizes right product quasigroups, and use this to nd axiom systems for right product (left, right) loops; in fact, we can obtain each of the latter by adjoining just one appropriate axiom to the right product quasigroup axiom system. We derive other properties of right product quasigroups and loops, and conclude by showing that the axioms for right product quasigroups are independent.
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].
Semigroups of Left Quotients—The Layered Approach
Communications in Algebra, 2004
A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be expressed as a ♯ b where a, b ∈ S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. Here a ♯ denotes the inverse of a in a subgroup of Q. We say that a left order S is straight in Q if in the above definition we can insist that a R b in Q. A complete characterisation of straight left orders in terms of embeddable *-pairs is available. In this paper we adopt a different approach, based on partial order decompositions of semigroups. Such decompositions include semilattice decompositions and decompositions of a semigroup into principal factors or principal *-factors. We determine when a semigroup that can be decomposed into straight left orders is itself a straight left order. This technique gives a unified approach to obtaining many of the early results on characterisations of straight left orders.
Corrections and Extensions in Left and Right Almost Semigroups
Punjab University Journal of Mathematics, 2021
In this paper we elaborated the concept that on what conditions left almost semigroup (LA-Semigroup), right almost semigroup (RA-Semigroup) and a groupoid become commutative and further extended these results on medial, LA-Group and RA-Group. We proved that the relation of LA-Semigroup with left double displacement semigroup (LDD-semigroup), RA-Semigroup with left double displacement semigroup (RDD-semigroup) is only commutative property. We highlighted the errors in the recently developed results on LA-Semigroup and semigroup [17, 1, 18] and proved that example discussed in [18] is semigroup with left identity but not paramedial. We extended results on locally associative LA-Semigroup explained in [20, 21] towards LA-Semigroup and RA-Semigroup with left zero and right zero respectively. We also discussed results on n-dimensional LA-Semigroup, n-dimensional RASemigroup, non commutative finite medials with three or more than three left or right identities and finite as well as infini...
Semigroups of left quotients—the uniqueness problem
Proceedings of the Edinburgh Mathematical Society, 1992
Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.The above result is then u...
On semilattices of semigroups and groups
In this paper, we have characterized a semigroup that is a semilattice of left (right) simple semigroups, a semigroup that is a semilattice of left (right) groups in terms of anti fuzzy left (right, two-sided) ideals, anti fuzzy (generalized) bi-ideals, anti fuzzy interior ideals and anti fuzzy quasi-ideals. Keywor
On right chain ordered semigroups
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
Semigroups with finitely generated universal left congruence
Monatshefte für Mathematik
We consider semigroups such that the universal left congruence ω is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that ω is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-FP 1 exactly when ω is finitely generated. Our investigations enable us to classify those semigroups such that ω is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that ω is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids. Keywords Monoids • Semigroups • Left congruences • Finitely generated • FP 1 • Pseudo-finite Communicated by J. S. Wilson.