The Smarandache Bryant Schneider Group Of Some Types Of Smarandache Loops (original) (raw)

Related papers

On Smarandache Bryant Schneider Group of A Smarandache Loop

… Combinatorics (International Book Series), Vol. 2, …

The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some properties of the Bryant Schneider Group of a loop are shown to be true for the Smarandache Bryant Schneider Group of a Smarandache loop. Some interesting and useful cardinality formulas are also established for a type of finite Smarandache loop.

The Smarandache Bryant Schneider Group Of A Smarandache Loop

The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some properties of the Bryant Schneider Group of a loop are shown to be true for the Smarandache Bryant Schneider Group of a Smarandache loop. Some interesting and useful cardinality formulas are also established for a type of finite Smarandache loop

A Pair of Smarandachely Isotopic Quasigroups and Loops Of The Same Variety

The isotopic invariance or universality of types and varieties of quasigroups and loops described by one or more equivalent identities has been of interest to researchers in loop theory in the recent past. A variety of quasigroups(loops) that are not universal have been found to be isotopic invariant relative to a special type of isotopism or the other. Presently, there are two outstanding open problems on universality of loops: semi automorphic inverse property loops(1999) and Osborn loops(2005). Smarandache isotopism(S-isotopism) was originally introduced by Vasantha Kandasamy in 2002. But in this work, the concept is re-restructured in order to make it more explorable. As a result of this, the theory of Smarandache isotopy inherits the open problems as highlighted above for isotopy. In this short note, the question 'Under what type of S-isotopism will a pair of S-quasigroups(S-loops) form any variety?' is answered by presenting a pair of specially S-isotopic S-quasigroups(loops) that both belong to the same variety of S-quasigroups(S-loops). This is important because pairs of specially S-isotopic S-quasigroups(e.g Smarandache cross inverse property quasigroups) that are of the same variety are useful for applications(e.g cryptography).

ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE

A Smarandache quasigroup(loop) is shown to be universal if all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f, g-principal isotopes are Smarandache f, g-principal isotopes. Conversely, it is shown that if these weak Smarandache loops of Bol-Moufang type are universal, then some autotopisms are true in the weak Smaran-dache sub-loops of the weak Smarandache loops of Bol-Moufang type relative to some Smarandache elements. Futhermore, a Smarandache left(right) inverse property loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes is shown to be universal if and only if it is a Smarandache left(right) Bol loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Also, it is established that a Smarandache inverse property loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes is universal if and only if it is a Smarandache Moufang loop in which all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Hence, some of the autotopisms earlier mentioned are found to be true in the Smarandache sub-loops of universal Smarandache: left(right) inverse property loops and inverse property loops.

AN HOLOMORPHIC STUDY OF THE SMARANDACHE CONCEPT IN LOOPS

If two loops are isomorphic, then it is shown that their holomorphs are also isomor-phic. Conversely, it is shown that if their holomorphs are isomorphic, then the loops are isotopic. It is shown that a loop is a Smarandache loop if and only if its holomorph is a Smarandache loop. This statement is also shown to be true for some weak Smaran-dache loops(inverse property, weak inverse property) but false for others(conjugacy closed, Bol, central, extra, Burn, A-, homogeneous) except if their holomorphs are nuclear or central. A necessary and sufficient condition for the Nuclear-holomorph of a Smarandache Bol loop to be a Smarandache Bruck loop is shown. Whence, it is found also to be a Smarandache Kikkawa loop if in addition the loop is a Smarandache A-loop with a centrum holomorph. Under this same necessary and sufficient condition, the Central-holomorph of a Smarandache A-loop is shown to be a Smarandache K-loop.

On certain isotopic maps of central loops

Proyecciones (Antofagasta), 2011

It is shown that the Holomorph of a C-loop is a C-loop if each element of the automorphism group of the loops is left nuclear. Condition under which an element of the Bryant-Schneider group of a C-loop will form an automorphism is established. It is proved that elements of the Bryant-Schneider group of a C-loop can be expressed a product of pseudo-automorphisms and right translations of elements of the nucleus of the loop. The Bryant-Schneider group of a C-loop is also shown to be a kind of generalized holomorph of the loop.

Smarandache Isotopy Theory Of Smarandache: Quasigroups And Loops

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.

Smarandache Isotopy Of Second Smarandache Bol Loops

The pair (G H , ·) is called a special loop if (G, ·) is a loop with an arbitrary subloop (H, ·) called its special subloop. A special loop (G H , ·) is called a second Smarandache Bol loop(S 2 nd BL) if and only if it obeys the second Smarandache Bol identity (xs·z)s = x(sz · s) for all x, z in G and s in H. The popularly known and well studied class of loops called Bol loops fall into this class and so S 2 nd BLs generalize Bol loops. The Smarandache isotopy of S 2 nd BLs is introduced and studied for the first time. It is shown that every Smarandache isotope(S-isotope) of a special loop is Smarandache isomorphic(S-isomorphic) to a S-principal isotope of the special loop. It is established that every special loop that is S-isotopic to a S 2 nd BL is itself a S 2 nd BL. A special loop is called a Smarandache G-special loop(SGS-loop) if and only if every special loop that is S-isotopic to it is S-isomorphic to it. A S 2 nd BL is shown to be a SGS-loop if and only if each element of its special subloop is a S 1 st companion for a S 1 st pseudo-automorphism of the S 2 nd BL. The results in this work generalize the results on the isotopy of Bol loops as can be found in the Ph.D. thesis of D. A. Robinson.

An Holomorphic Study Of Smarandache Automorphic and Cross Inverse Property Loops

By studying the holomorphic structure of automorphic inverse property quasigroups and loops[AIPQ and (AIPL)] and cross inverse property quasigroups and loops[CIPQ and (CIPL)], it is established that the holomorph of a loop is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop if and only if its Smarandache automor-phism group is trivial and the loop is itself is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop.