Principal congruence properties of some algebras with pseudocomplementation.

*(English)*Zbl 0801.06023A \(p\)-algebra (i.e., a pseudocomplemented bounded lattice) \(L\) is said to be quasi-modular, if it satisfies the identity \(((x\wedge y)\vee z^{**})\wedge x=(x\wedge y)\vee (z^{**}\wedge x)\). A \(p\)-algebra endowed with a dual pseudocomplementation is called a double \(p\)-algebra. Any lattice-based algebra is said to have the principal join (intersection), abbreviated PJP (PIP), if the join (intersection) of any pair of principal congruences is again principal.

Theorems 3.4 and 3.8 of this paper characterize the algebras with the PJP within the classes of quasi-modular \(p\)-algebras and of distributive \(p\)- algebras, respectively. Theorem 3.10 establishes that any distributive \(p\)-algebra having the PJP has the PIP, while Theorem 4.7 characterizes the algebras with the PIP within the class of distributive double \(p\)- algebras having the PJP. Theorem 4.2 characterizes the distributive double \(p\)-algebras with the principal congruence property, i.e. such that every principal congruence of \(L\) is a principal congruence of the lattice reduct of the algebra. There are also numerous lemmas and corollaries that have an intrinsic interest.

Theorems 3.4 and 3.8 of this paper characterize the algebras with the PJP within the classes of quasi-modular \(p\)-algebras and of distributive \(p\)- algebras, respectively. Theorem 3.10 establishes that any distributive \(p\)-algebra having the PJP has the PIP, while Theorem 4.7 characterizes the algebras with the PIP within the class of distributive double \(p\)- algebras having the PJP. Theorem 4.2 characterizes the distributive double \(p\)-algebras with the principal congruence property, i.e. such that every principal congruence of \(L\) is a principal congruence of the lattice reduct of the algebra. There are also numerous lemmas and corollaries that have an intrinsic interest.

Reviewer: S.Rudeanu (Bucureşti)

##### MSC:

06D15 | Pseudocomplemented lattices |