We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones C1, C2, C3 such that: (1) the interval [C1, O] in the lattice of local clones is (as a lattice) isomorphic to {0, 1, 2, . . .} under the divisibility relation, (2) the interval [C2, O] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilat...

Suppose R is ring with 1, and HLe(R) denotes the variety of modular lattices generated by the class of lattices of submodules of all R-modules. An algorithm using Mal'cev conditions is given for constructing integers m > 0 and n -> 1 from any given lattice polynomial inclusion formula d ~ e. The main result is that d c e is satisfied in every lattice in H~(R) if and only if there exists x in R ...

The concept of an MV-algebra was introduced by Chang [4] as an algebraic basis for many-valued logic. It turned out that MV-algebras are a subclass of a more general class of effect algebras [7, 6]. Namely, MV-algebras are in one-to-one correspondence with lattice ordered effect algebras satisfying the Riesz decomposition property [2], the latter are called MV-effect algebras. In the study of c...

⊇ Basic ⊇ Congruence ⊇ Division of sets of intervals, those sets closed under taking (a) similar intervals, (b) sub-intervals, (c) abutting intervals, (d) suprema, respectively. We let B(Λ) and D(Λ) be the family of all basic sets and division sets, respectively. Abstract and basic sets are not particularly important, but the do help with the organization. However, observe that the family B(Λ) ...

Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the correspondin...