The Algebraic Sets of Broad Algebras

Abstract

The paper is devoted to the questions of algebraic geometry of universal algebras, more precisely, to the structures of algebraic sets of this algebras. It is introduced the concept of broad universal algebra. Some natural examples of such universal algebras are given including the lattices of functional clones on sets, the groups of permutations on sets, the lattices of partitions on sets, the countable free Boolean algebras, and the direct powers of universal algebras and others. Some special features of the structures of algebraic sets of broad universal algebras are considered. We prove the algebraic $n$-completnes of broad universal algebras. The results on the structure of the quasiorder which is generated on the broad universal algebra by its inner homomorphisms (homomorphisms between subalgebras) are presented. Some estimations of the powers of algebraic sets of the broad universal algebras are given. Some results on the minimal sets which are generated of algebraic sets of the broad universal algebras are obtained.

Keywords

• algebraic set
• broad algebra
• $n$-completeness algebra