Strongly Algebraically Closed Lattices in ℓ-groups and Semilattices

Abstract

Systematic study of universal algebraic geometry is done in a series of articles by V. Remeslennikov, A. Myasnikov and E. Daniyarova in ( [2–5]). In [12] J. Schmid studied algebraically closed and existentially closed distributive lattices. He proved that any Boolean lattice is algebraically closed. J. Schmid asked about the situation in which a distributive lattice is strongly algebraically closed. In [10] we defined the notion of strongly algebraically closed lattices and proved that if such a lattice is also complete Boolean and q′-compact, then it is strongly algebraically closed. The current paper continues the study of [10]. We suggest [7–9], [11], and [13] in order to give a precise definition of the notion algebraically closedness of model theory and Boolean algebras. In Section 1, it is proved that if the lattice of the א0-classes Cl(G) of a full l-group G is complete and q′-compact, then Cl(G) is a strongly algebraically closed lattice. Also, we prove that, if the set of polars P(G) of an l-group G is q′-compact, then P(G) is a strongly algebraically closed lattice. At the end of this section, we will show that, if the complemented l-ideals of a complete l-group is q′-compact, then it is a strongly algebraically closed lattice. In the final section of this paper we study some interesting applications of strongly algebraically closed lattices about pseudocomplemented semilattices and the set of invariant elements of a dimension ortholattice.