On Strongly Algebraically Closed Lattices

Abstract

In this article some fundamental properties of existentially and algebraically closed lattices are investigated. We also define the notion of strongly algebraically closed lattices and we show that a q ′ -compact complete boolean lattice is strongly algebraically closed.

Strongly Algebraically Closed Lattices in ℓ-groups and Semilattices

“…We prove Let x be an element of the ℓ-group G. Recall that the positive part, negative part, and absolute value of x are defined by: On the other hand, we know that the Boolean algebra of polars of an ℓ-group is complete, and this property is reflected in its dual space and is an essential ingredient for the representation of an archimedean ℓ-group. So, if G is an ℓ-group, then the set of polars of a G is complete Boolean algebra, Now, by ( [10], Theorem 3.5), completes the proof.…”

“…A lattice L is called algebraically closed, if any finite consistent system of equations with coefficients from L, has a solution in L. A system S with coefficients in L is called consistent, if there is an extension K of L, such that S has a solution in K. One can generalize this definition to an arbitrary class of lattices. A lattice L in a class X is said to be strongly algebraically closed if every system (not necessarily finite) of equations with parameters in L which has a solution in some extension K ∈ X, has already a solution in L (see [10] and [12], for more details).…”

“…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].…”

“…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][8][9], [11], and [13] in order to give a precise definition of the notion algebraically closedness of model theory and Boolean algebras.…”

“…Since any intersection of closed ℓ-ideals is closed. We will have that the complemented ℓ-ideals of any complete ℓ-group form a complete Boolean algebra, hence the complemented ℓ-ideals of G is q ′ -compact and so by applying that ( [10], Theorem 3.5), the complemented ℓ-ideals of any complete ℓ-group is a strongly algebraically closed lattice.…”