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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.