Gradimir Vojvodic

L -fuzzy sets and codes

Abstract A decomposition of an L -valued finite fuzzy set ( L is a lattice) gives a family of characteristic functions, which can be considered as a binary block-code. Using a previous theorem of synthesis for fuzzy sets, we give conditions under which an arbitrary block-code corresponds to an L -valued fuzzy set. An explicit description of the Hamming distance, as well as of any code distance is also given, all in lattice-theoretic terms. Finally, we give necessary and sufficient conditions under which a linear code corresponds to an L -valued fuzzy set. It turns out that in such case the lattice L has to be Boolean.

On the lattice of weak congruence relations

We consider the setCw(A) of weak congruences on an algebraA, i.e. of all symmetric and transitive subalgebras ofA ×A. (Some other generalizations of compatible relations were given in [2] and [6]).Cw(A) coincides with the set of all congruences on all subalgebras ofA.We prove here that (iCw(A),⩽) is an algebraic lattice having as a sublattice the lattice of all congruences onA, and as a retract the lattice of all its subalgebras. In the second part we give necessary and sufficient conditions (one is the CEP) for modularity as well as for distributivity of the lattice of weak congruences.

Algebras of fuzzy sets

In this paper we investigate two kinds of algebras of fuzzy sets, which are obtained by using Zadehs extension principle. We give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets. We prove that if the structure of truth values is a complete residuated lattice, the induced algebra of a subalgebra of an algebra A can be embedded into the induced algebra of fuzzy sets of A. For direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product. In the case of homomorphisms and direct products, the two kinds of algebras of fuzzy sets behave in different ways.

Minimal partial hyperclones on a two-element set

Let A be a two-element set. It has been proven by Machida that the lattice of hyperclones has continuum cardinality. The current paper first determines all the minimal partial hyperclones. Following this, the authors give all three-element subsets of minimal hyperoperations whose union generates the clone of all hyperoperations as well as all four minimal subsets of minimal partial hyperoperations whose union generates the clone of all partial hyperoperations.

On the partial hyperclone lattice

For any finite set A, the partial clone lattice on A is embedded into the partial hyperclone lattice on A. It is shown that there are maximal intervals in the partial hyperclone lattice and there are four minimal partial hyperclones such that their join contains all partial hyperoperations. It is proved in (T. Drescher et al., 2001) that the mapping /spl lambda/ from the lattice of partial hyperclones on A into the lattice of clones of operations on P(A) defined by /spl lambda/(C)=/spl delta/(C/sup #/), where /spl delta/(C/sup #/) is the clone of operations on P(A) generated by C/sup #/, is an order embedding, but not a full one. In this paper, it is proved that there are continuum many clones of operations on P(A) that are in the interval [/spl lambda/(J/sub A/), /spl lambda/(Hp/sub A/)] but these are not in the set im/spl lambda/ of all images of the mapping /spl lambda/, where J/sub A/ is the set of all (partial) hyperprojections and Hp/sub A/ is the set of all partial hyperoperations on A.

Commuting Hyperoperations

Centralizer of a set of hyperoperations F is a clone of hyperoperations that commute with all hyperoperations from F. There are several ways to define this commuting operator which imply several definitions of centralizers of sets of hyperoperations and they are considered in this paper. In order to obtain their properties, we discuss the definition of graph of hyperoperation, relation on A^n × (P(A) \ {0}) and lifting of such relation to (P(A) \ {0}^n+1.

Relative completeness with respect to two unary functions

Abstract One of the most important results is the fact that the number of clones is a continuum for k ⩾3, while the corresponding set for k =2 is countable. This shows a sharp difference when we go from the binary to the ternary case. This paper discusses the relative completeness with respect to the clone generated by two unary functions and show the sharp difference when we go from four-valued logic to k -valued logic for k >4, as well. The number of maximal clones over a finite set is finite and increases when k increases. However, there are two relative maximal clones if k =3,4 and there is one relative maximal clone if k >4.