Semirings satisfying properties of distributive type
Abstract: Any distributive lattice admits a semiring structure in a natural way and this particular semiring satisfies some new distributive properties. The purpose of this note is to study whether a semiring with some or all of these new distributive properties admits the natural distributive lattice structure, and thus obtain Birkhoff's theorem as a corollary. Furthermore, characterizations of semirings satisfying some or all of these new distributive properties are obtained.
-  Garrett Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, revised edition, American Mathematical Society, New York, N. Y., 1948. MR 0029876
-  Mario Petrich, Introduction to semigroups, Charles E. Merrill Publishing Co., Columbus, Ohio, 1973. Merrill Research and Lecture Series. MR 0393206
Keywords: Band, rectangular band, left zero semigroup, distributive lattice
Article copyright: © Copyright 1981 American Mathematical Society