156 research outputs found
Equivalence of operations with respect to discriminator clones
For each clone C on a set A there is an associated equivalence relation,
called C-equivalence, on the set of all operations on A, which relates two
operations iff each one is a substitution instance of the other using
operations from C. In this paper we prove that if C is a discriminator clone on
a finite set, then there are only finitely many C-equivalence classes.
Moreover, we show that the smallest discriminator clone is minimal with respect
to this finiteness property. For discriminator clones of Boolean functions we
explicitly describe the associated equivalence relations.Comment: 17 page
A note on minors determined by clones of semilattices
The C-minor partial orders determined by the clones generated by a
semilattice operation (and possibly the constant operations corresponding to
its identity or zero elements) are shown to satisfy the descending chain
condition.Comment: 6 pages, proofs improved, introduction and references adde
Clones with finitely many relative R-classes
For each clone C on a set A there is an associated equivalence relation
analogous to Green's R-relation, which relates two operations on A iff each one
is a substitution instance of the other using operations from C. We study the
clones for which there are only finitely many relative R-classes.Comment: 41 pages; proofs improved, examples adde
Partially ordered pattern algebras
A partial order ≤ on a set A induces a partition of each power An into "patterns" in a natural way. An operation on A is called a ≤-pattern operation if its restriction to each pattern is a projection. We examine functional completeness of algebras with ≤-pattern fundamental operations
Clones with finitely many relative R-classes
For each clone C on a set A there is an associated equivalence relation
analogous to Green's R-relation, which relates two operations on A iff each one
is a substitution instance of the other using operations from C. We study the
clones for which there are only finitely many relative R-classes.Comment: 41 pages; proofs improved, examples adde
The arity gap of polynomial functions over bounded distributive lattices
Let A and B be arbitrary sets with at least two elements. The arity gap of a
function f: A^n \to B is the minimum decrease in its essential arity when
essential arguments of f are identified. In this paper we study the arity gap
of polynomial functions over bounded distributive lattices and present a
complete classification of such functions in terms of their arity gap. To this
extent, we present a characterization of the essential arguments of polynomial
functions, which we then use to show that almost all lattice polynomial
functions have arity gap 1, with the exception of truncated median functions,
whose arity gap is 2.Comment: 7 page
- …