Article
Keywords:
modes; Szendrei modes; subreducts; semimodules; equational theory
Summary:
We prove a theorem describing the equational theory of all modes of a fixed type. We use this result to show that a free mode with at least one basic operation of arity at least three, over a set of cardinality at least two, does not satisfy identities selected by 'A. Szendrei in {\it Identities satisfied by convex linear forms\/}, Algebra Universalis {\bf 12} (1981), 103--122, that hold in any subreduct of a semimodule over a commutative semiring. This gives a negative answer to the question raised by A. Romanowska: Is it true that each mode is a subreduct of some semimodule over a commutative semiring?
References:
[3] Golan J.S.:
The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Longman Scientific & Technical, Harlow, 1992.
MR 1163371 |
Zbl 0780.16036
[4] Ježek J., Kepka T.:
Medial Groupoids. Rozpravy ČSAV 93/2, Academia, Praha, 1983.
MR 0734873
[5] Ježek J., Kepka T.:
Linear equational theories and semimodule representations. Internat. J. Algebra Comput. 8 (1998), 599-615.
MR 1675018
[8] Szendrei À.:
Identities satisfied by convex linear forms. Algebra Universalis 12 (1981), 103-122.
MR 0608653 |
Zbl 0458.08006