Showing papers on "Free algebra published in 1969"

A Remark on Varieties of Lattices and Semigroups

Abstract: We consider for any variety V, the lattice L(V) of subvarieties of V. It is known (H. Neumann [2 ]) that if V is the variety of groups, then V is not the join of any finite number of proper subvarieties. Recently, B. Jonsson [1 ] has proved a similar result for the variety of lattices. His proof though short requires rather sophisticated machinery. In this remark we show by elementary methods that for any two lattice identities there is a nontrivial identity which each implies independently. A similar approach also gives the corresponding result for semigroups. A trivial observation used in the lemmas is that h1= h2 is a nontrivial identity if and only if hi and h2 represent distinct elements of a free algebra. From these two lemmas it follows easily that for neither lattices nor semigroups is the variety a join of a finite number of subvarieties.

Subalgebras of free algebras of some varieties of multioperator algebras

Abstract: The problem whether subalgebras of free algebras of various varieties are free plays an important role in general algebra. For some varieties of linear algebras over a field the problem was solved by Kurosh [1] and Shirshov [2], [3]. Kurosh [4] introduced the concept of multioperator algebra over a field and proved that every subalgebra of a free multioperator algebra is free. This paper is devoted to a study of varieties of multioperator algebras given by identities of a special form; particular cases are the commutative and anticommutative laws for classical linear algebras. The main result of the paper comprises the freeness theorem mentioned above for subalgebras of a free multioperator algebra, as well as parallel theorems in Shirshov's papers [2] on the freeness of subalgebras of a free commutative and a free anticommutative algebra; the methods of this last article are maintained without essential modifications.