Adam Mata
supervisor: Anna Zamojska-Dzienio
A semilattice is a partially ordered set (`S,≤`) in which for any non-empty finite subset its meet (greatest lower bound) exists. Algebraically it can be described as the structure (`S,∙`) where:
`x ∙ (y ∙ z) = (x ∙ y) ∙ z`,
`x ∙ y = y ∙ x`,
`x∙x = x`,
for all `x, y, z` in `S`. Both approaches are related by the rule: `x≤y⇔x∙y=x`.
A Boolean semilattice is a Boolean algebra with additional semilattice operation `∙`. This is an example of a Boolean algebra with operators which are widely applied in modal logic.
An algebraic variety is a collection of structures defined by equalities (identities). A class of all Boolean semilattices is a variety.
We consider a theory of a particular variety as all statements about structures in the variety (formulas) which are true.
Let us denote the theory of some variety `V` as `Th(V)`. The variety `V` is called decidable if there exists an algorithm which for any given statement `Ф` computes within finite steps, the answer for the question whether `Ф` belongs to `Th(V)` or not.
The main hypothesis of my work is that the variety of all Boolean semilattices is undecidable although there may exist some significant subclasses of this variety which may turn out to be decidable.