Vol. 36, No. 1, 1971

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 331: 1
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Topisms and induced non-associative systems

Kenneth Paul Baclawski and Kenneth Kapp

Vol. 36 (1971), No. 1, 45–54
Abstract

A triple of bijections (γ;α,β) from one binary system onto another is called an isotopism if the maps transfer the multiplication: (xy)γ = xαyβ. These maps have been used by Albert, Bruck and others in the study of quasigroups and loops. A natural generalization of isotopism is given to a triple of maps called a topism, which transfers just the multiplication, as above, from one binary system into another. It will be shown, (1.7), that a topism from one quasigroup into another is an isotopism if and only if any one of the maps is one-to-one and onto.

Two maps, α,β on a binary system (A,) induce a new binary operation, , defined in the natural fashion x y = xαyβ. The relationship between topic imbeddings and induced systems is studied in §2. It is shown, (2.3), that one groupoid is topically imbeddable in another precisely when it is isomorphic to a subgroupoid of an induced groupoid of the second. Thus, (2.7), two quasigroups are isotopic if and only if one is isomorphic to an induced groupoid of the other.

Finally, the imbeddings of nonassociative binary systems into semigroups and groups are considered. It is shown, (3.1) and (3.2), that groupoids can be imbedded as ideals in semigroups. However, it is seen (3.4) that a groupoid with identity is topically imbeddable in a group precisely when the groupoid is isomorphic to a subsemigroup, with identity, of the given group. From this a generalization, (3.6), of Albert’s Theorem that a loop is isotopic to a group if and only if they are isomorphic is obtained.

Mathematical Subject Classification
Primary: 20.95
Milestones
Received: 25 February 1970
Published: 1 January 1971
Authors
Kenneth Paul Baclawski
Kenneth Kapp