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Generalized semialgebras over semirings

1988; Springer Nature; Linguagem: Inglês

10.1007/bfb0083446

1617-9692

Hanns Joachim Weinert,

Natural Language Processing Techniques

For an associative ring S, the classical concept of an algebra A over S forces S to be commutative in nearly all cases of some interest. E. g., polynomial rings or matrix rings over a non-commutative ring S with identity are not classical algebras over S. For this reason, H. Zassenhaus [21] and G. Pickert [13] have introduced a more general concept of an algebra A over S. If A has an infinite basis over S, both concepts can be generalized in another direction, which may be illustrated by rings of formal power series. At third, certain semirings A constructed from semirings S have become important tools e.g. in the theory of automata and formal languages, in particular again those in which also infinite sums occur. The purpose of this paper is to deal with all these different concepts in a unique way as special cases of generalized semialgebras over semirings. Since we have to describe all the material we want to combine, some parts of this paper have the character of a survey article.