Remarks on endolinear algebras and endoalgebras
by
Kazimierz Glazek
University of Zielona Góra, Poland

Recall that the class of quasi-linear algebras is equivalent to the class of separable variable algebras (see [1] and [5]). Linear spaces and modules over arbitrary rings are examples of such algebras. If an algebra A = (A; F) is a quasilinear algebra, then A is a subset of an abelian group (G; +) and 0 is a constant term operation of A. We assume that the group operation + is a term operation of A and 0 is only constant term operation of A. In this case A is called an endolinear algebra, because every n-ary term operation is a sum of summands of the form f_i(x_i), where f_i is an endomorphism of the semigroup (A; +). Any endolinear algebra A is a TC-algebra, and the set of all unary term operations of A forms a semiring. Moreover, A is a so-called endoalgebra of the monoid (A; +, 0). See [3] and [4]. We investigate such kinds of algebras. For examples, we have some results for Q-independent subsets of such algebras (with respect to specified families Q of mappings).

References:

[1] S. Fajtlowicz, K. Gazek, and K. Urbanik, Separable variables algebras, Colloq. Math. 19 (1966), 161-171.

[2] K. Gazek, Independence with respect to a family of mappings in abstract algebras, Dissertationes Math. 81 (1971), 1-55.

[3] K. Gazek, Algebras of Algebraic Operations and Morphisms of Algebraic Systems (in Polish), Acta Universitatis Wratislaviensis no. 1602 (1994), 148 pp.

[4] K. Gazek, Morphisms of general algebras without fixed fundamental operations, Contemporary Math. 184 (1995), 117-137.

[5] A. Hulanicki, E. Marczewski, and J. Mycielski, Exchange of independent sets in abstract algebras. I, Colloq. Math. 14 (1966), 203-215.

Date received: May 24, 2004