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Commutation of operations and its relationship with Menger and Mann superpositions

Fedir M. Sokhatsky (2004)

Discussiones Mathematicae - General Algebra and Applications

The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.

Commutative subloop-free loops

Martin Beaudry, Louis Marchand (2011)

Commentationes Mathematicae Universitatis Carolinae

We describe, in a constructive way, a family of commutative loops of odd order, n ≥ 7 , which have no nontrivial subloops and whose multiplication group is isomorphic to the alternating group 𝒜 n .

Commutators and associators in Catalan loops

Jan M. Raasch (2010)

Commentationes Mathematicae Universitatis Carolinae

Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.

Completely dissociative groupoids

Milton Braitt, David Hobby, Donald Silberger (2012)

Mathematica Bohemica

In a groupoid, consider arbitrarily parenthesized expressions on the k variables x 0 , x 1 , ⋯ x k - 1 where each x i appears once and all variables appear in order of their indices. We call these expressions k -ary formal products, and denote the set containing all of them by F σ ( k ) . If u , v ∈ F σ ( k ) are distinct, the statement that u and v are equal for all values of x 0 , x 1 , ⋯ x k - 1 is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds...

Congruences and ideals in ternary rings

Ivan Chajda, Radomír Halaš, František Machala (1997)

Czechoslovak Mathematical Journal

A ternary ring is an algebraic structure ℛ = ( R ; t , 0 , 1 ) of type ( 3 , 0 , 0 ) satisfying the identities t ( 0 , x , y ) = y = t ( x , 0 , y ) and t ( 1 , x , 0 ) = x = ( x , 1 , 0 ) where, moreover, for any a , b , c ∈ R there exists a unique d ∈ R with t ( a , b , d ) = c . A congruence θ on ℛ is called normal if ℛ / θ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on ℛ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

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