arXiv subjects

Muhammad Shah

Publications and source records attributed to Muhammad Shah.

11 recordsLinked to original sources

arXiv2026

A Simple Way of Getting Large Examples of Osborn Loops

Examples and counterexamples define the boundaries of mathematical propositions, test foundational conjectures, and resolve open structural problems in abstract algebra. In loop theory, an Osborn loop is a crucial generalization of a Moufang loop satisfying a specialized variable-sandwiching identity. While small finite non-associative examples are known, constructing large examples of proper Osborn loops (loops that are neither conjugacy closed nor Moufang) remains a computational bottleneck. In this paper, we establish an efficient framework for generating large proper Osborn loops by taking the direct product of non-associative conjugacy closed (CC) loops and Moufang loops. We outline the boundary constraints necessary to prevent structural collapse into sub-varieties and provide explicit examples up to order 2025 validated via the GAP package \texttt{LOOPS}.

math.RA↗

arXiv2026

Enumerating AG-monoids algebraically

An AG-monoid is an AG-groupoid (a groupoid satisfying the identity called left invertive law $(xy)z=(zy)x$) and having a left identiy. In this paper we enumerate AG-monoids algebraically and then implement them in GAP to compute them computationally.

math.GR↗

arXiv2015

On cyclic associative Abel-Grassman groupoids

A new subclass of AG-groupoids, so called, cyclic associative Abel-Grassman groupoids or CA-AG-groupoid is studied. These have been enumerated up to order $6$. A test for the verification of cyclic associativity for an arbitrary AG-groupoid has been introduced. Various properties of CA-AG-groupoids have been studied. Relationship among CA-AG-groupoids and other subclasses of AG-groupoids is investigated. It is shown that the subclass of CA-AG-groupoid is different from that of the AG{*}-groupoid as well as AG{*}{*}-groupoids.

math.GR↗

arXiv2014

Enumeration of Bi-commutative AG-groupoids

A groupoid satisfying the left invertive law: $ab\cdot c=cb\cdot a$ is called an AG-groupoid and is a generalization of commutative semigroups. We consider the concept of bi-commutativity in AG-groupoids and thus introduce left commutative AG-groupoids, right commutative AG-groupoids and bi-commutative AG-groupoids.

math.GR↗

arXiv2014

Some General Properties of Stein-AG-groupoids and Stein-AG-Test

A groupoid that satisfying the left invertive law is called an AG-groupoid.this concept is extended to introduce a Stein AG-groupoid. We provethe existence by providing some non-associative examples. We also explore some basic and general properties of these AG-groupoids and find their relations with other subclasses of AG-groupoids.

math.GR↗

arXiv2014

Some General Properties of LAD and RAD AG-groupoids

A groupoid that satisfies the left invertive law: $ab\cdot c=cb\cdot a$ is called an AG-groupoid. We extend the concept of left abelian distributive groupoid (LAD) and right abelian distributive groupoid (RAD) to introduce new subclasses of AG-groupoid, left abelian distributive AG-groupoid and right abelian distributive AG-groupoid. We give their enumeration up to order 6 and find some basic relations of these new classes with other known subclasses of AG-groupoids and other relevant algebraic structures. We establish a method to test an arbitrary AG-groupoid for these classes.

math.GR↗

arXiv2014

Fuzzy Cosets and Quotient Fuzzy AG-subgroups

In this paper we extend the concept of fuzzy AG-subgroups. We introduce some results in normal fuzzy AG-subgroups. We define fuzzy cosets and quotient fuzzy AG-subgroups, and prove that the sets of their collection form an AG-subgroup and fuzzy AG-subgroup respectively. We also introduce the fuzzy Lagrange's Theorem of AG-subgroup. It is known that the condition $\mu(xy)=\mu(yx)$ holds for all $x,y$ in fuzzy subgroups if $\mu$ is normal, but in fuzzy AG-subgroup we show that it holds without normality.

math.GM↗

arXiv2014

On Modulo AG-groupoids

A groupoid G is called an AG-groupoid if it satisfies the left invertive law: (ab)c = (cb)a. An AG-group G, is an AG-groupoid with left identity e \in G (that is, ea = a for all a \in G) and for all a \in G there exists a' \in G such that a.a' = a'.a = e. In this article we introduce the concept of AG-groupoids (mod n) and AG-group (mod n) using Vasantha's constructions [1]. This enables us to prove that AG-groupoids (mod n) and AG-groups (mod n) exist for every integer n \geq 3. We also give some nice characterizations of some classes of AG-groupoids in terms of AG-groupoids (mod n).

math.GR↗

arXiv2014

The Multiplication Group of an AG-group

We investigate the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: the multiplication group of an AG-group of order n is non-abelian group of order 2n and its left section is an abelian group of order n. The inner mapping group of an AG-group of any order is a cyclic group of order 2.

math.GR↗

arXiv2014

Left Transitive AG-groupoids

An AG-groupoid is an algebraic structure that satisfies the left invertive law: (ab)c =(cb)a. We prove that the class of left transitive AG-groupoids (AG-groupoids satisfying the identity, ab.ac = bc) coincides with the class of T2-AG-groupoids. We also develop a simple procedure to test whether an arbitrary groupoid is left transitive AG-groupoid or not. Further we prove that, (i). Every left transitive AG-groupoid is transitively commutative AG-groupoid (ii) For left transitive AG-groupoid the properties of flexibility, right alternativity, AG*, right nuclear square, middle nuclear square and commutative semigroup are equivalent.

math.GR↗

arXiv2011

AG-groups and other classes of right Bol quasigroups

By a result of Sharma, right Bol quasigroups are obtainable from right Bol loops via an involutive automorphism. We prove that the class of AG-groups, introduced by Kamran, is obtained via the same construction from abelian groups. We further introduce a new class of Bol* quasigroups, which turns out to correspond, as above, to the class of groups. Sharma's correspondence allows an efficient implementation and we present some enumeration results for the above three classes.

math.GR↗