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Left symmetric left distributive magmas and hypersubstitutions

Vanzurova A.

Demonstratio Mathematica

2006

Vol. 39, nr 3

491-506

Our aim is to generalize results reached in [A-D 98] and [V 04]. In [A-D 98], normal forms for terms with respect to the variety SIE of right symmetric idempotent entropic magmas are used to derive multiplication in the magma of normal form hypersubstitutions with respect to SIE, the monoid of SIE-proper normal form hypersubstitutions is found, and hyperidentities are discussed. In [V 04], a similar project is solved for the variety SID of left symmetric left distributive idempotent magmas (in which the variety dual to SIE is contained as a subvariety). Droppping idempotency we obtain a generalization, the variety SD of left symmetric left distributive magmas. We use again (naturally arising) normal forms for terms in SD to study the magma of SD-normal form hypersubstitutions, its multiplication (with six idempotents), and describe the monoid of SD-proper normal form hypersubstitutions. Comparison with the previous cases might be interesting.

On a general construction of diagonal algebras

Klouda J., Vanzurova A.

Demonstratio Mathematica

2000

Vol. 33, nr 2

223-230

In [4, 5], J. Plonka introduced a concept of the diagonal n-dimensional algebra as well as the generalized n-dimensional diagonal algebra as a generalization of a certain class of semigroups. In [5], (generalized) n-dimensional diagonal algebras were characterized as diagonal algebras (without idempotency property) (A U D, p) such that A, D are disjoint, the subalgebra (D,p|D) is an n-dimensional diagonal idempotent algebra, and there exists a retraction f : AU-D - D satisfying p(x1,..., xn) = p(f(x1), .......,f(xn). The theory was further developed and generalized for universal algebras of arbitrary types, e.g. by J. Ślapal who applied a categorical view point. In [9] a generalized (i.e. without idempotency property) diagonal algebra of type I is introduced, and the necessary and sufficient condition for diagonality is given. In the case of p being a binary operation it is proved that diagonality of a grupoid is equivalent with the condition x (y z) = x z = (xy)z so that diagonal grupoids are semigroups. Further, a complete survey of diagonal grupoids with cardinality |X| = 2,3 is given, and an example of a diagonal grupoid with |X| = 12 is presented. In [13] all diagonal and idempotent binary operations on a set consisting of four elements are presented. In [10] it is shown that the class of diagonal algebras of a fixed type together with the family of homomorphisms as morphisms form a cartesian closed category. In the present paper a general construction of diagonal algebras is derived and it is proved that any diagonal algebra in the sense of [9] can be created in this way. The examples show how the construction enables us e.g. to classify generalized diagonal binary algebras on a finite set, and a computer program constructed by V. Tichy which describes representatives of classes (with respect to isomorphism and order of variables) of binary and ternary diagonal algebras of orders up to twenty can be found on http://kag.upol.cz/katedra/vanzurov.html under the name AlgeAl.