Latin squars, -Quasigroups and graph decompositions
Left MQQs whose left parastrophe is also quadratic
A left quasigroup of order that can be represented as a vector of Boolean functions of degree 2 is called a left multivariate quadratic quasigroup (LMQQ). For a given LMQQ there exists a left parastrophe operation defined by: that also defines a left multivariate quasigroup. However, in general, is not quadratic. Even more, representing it in a symbolic form may require exponential time and space. In this work we investigate the problem of finding a subclass of LMQQs whose left parastrophe...
Lexicographic product decompositions of half linearly ordered loops
In this paper we prove for an hl-loop an assertion analogous to the result of Jakubík concerning lexicographic products of half linearly ordered groups. We found conditions under which any two lexicographic product decompositions of an hl-loop with a finite number of lexicographic factors have isomorphic refinements.
Linear identities in graph algebras
We find the basis of all linear identities which are true in the variety of entropic graph algebras. We apply it to describe the lattice of all subvarieties of power entropic graph algebras.
Logarithmetics and quasigroup structure
Loop characters
A survey of the basic results of loop characters is given on the lines of the treatment of the author and J.D.H. Smith for characters of quasigroups, including some recent deveploments. One of the successes of the theory has been its suggestive influence on the theory of association schemes, group representations and the theory of the group determinant, and selected results arising are described. A section is devoted to an explanation of how the tool of loop characters has not yet been as startlingly...
Loop cohomology
Loops and quasigroups: Aspects of current work and prospects for the future
This paper gives a brief survey of certain recently developing aspects of the study of loops and quasigroups, focussing on some of the areas that appear to exhibit the best prospects for subsequent research and for applications both inside and outside mathematics.
Loops embedded in generalized Cayley algebras of dimension , .
Loops which are cyclic extensions of their nuclei
Loops whose translations generate the alternating group