On distributive quad-lattices

Duality for distributive bisemilattices

“…A bisemilattice {S, •, +) satisfying the equations x -( y + z) = x -y + x -z , x + y • z = (x + y ) • (x + z ) is called a distributive bisemilattice (see [15] and [17], where distributive bisemilattices are investigated under the name of distributive quasilattice, and [22]). Examples for distributive bisemilattices are distributive lattices and the bisemilattices of the form (S, •, •) obtained from a semilattice (S,-).…”

“…Every left normal band may be decomposed as a Plonka sum into the disjoint union of its maximal left-zero subbands according to the following theorem (cf. [9], [15], [22]).…”

“…The operation * is also a partition operation for (S, •, +) in order to obtain this bisemilattice as a Plonka sum of its maximal distributive sublattice (cf. [14], [15], [22]). In the sequel we call (S, •,+) in this context the Plonka sum of the distributive lattices (S t , -, +) over the semilattice (/, •) by the homomorphisms (P it j)jj and denote the elements of S t by a f , b { , etc.…”

Algebraic Analysis of Demodalised Analytic Implication

“…Distributive bisemilattices were introduced by Płonka under the name of distributive quasilattices in [21] as a substantive application of the techniques reviewed in the previous subsection. Distributive bisemilattices are the algebras that are representable as Płonka sums over direct systems of distributive lattices and include as limit cases both distributive lattices and semilattices.…”

Hyperidentities of some generalizations of lattices

“…Applying results of [24], [17] it is easy to see that the set _ I DNR of the identities (N1), (C1), (L3), (L12), their duals and x+x: x · x is a base for N(R(D)). Clearly, …”