Narrow Results

Idempotency languages are generated from a single word by iterated application of rules of the form $u^m\to u^n$ for natural numbers m and n. We investigate these languages over alphabets of only one or two letters. The conditions under which the underlying rewrite relations are confluent are fully characterized. Then for many combinations of the parameters m and n we answer the question, whether the corresponding idempotency languages are regular or not. What remains open are only the cases where $2\le m<n$.

We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\dots ,x)=x$ for all $x\in X$. We first provide a structure theorem for these symmetric $n$-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong $n$-ary semilattice of $n$-ary semigroups and we show that the symmetric $n$-ary bands are exactly the strong $n$-ary semilattices of $n$-ary extensions of Abelian groups whose exponents divide $n-1$. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric $n$-ary band to be reducible to a semigroup.

This paper deals with the problem of establishing a new class of aggregation operators, which are close to the averaging, without being necessarily idempotent. An axiomatic approach is followed through the definition of crucial properties which characterize these aggregators, the mathematical consequences are illustrated and many examples that extend some important classical averaging operators are shown.

This paper deals with aggregation operators. A new form of idempotency, called asymptotic idempotency, is introduced. A critical discussion of the basic notion of aggregation operator, strictly connected with asymptotic idempotency, is provided. Some general construction methods of symmetric, asymptotically idempotent aggregation operators admitting a neutral element are illustrated.

We give the general form of an idempotent, associative, nondecreasing and continuous binary aggregation operation in a connected order topological space. The particular case of the unit interval is studied and the choice of weights is also analized. Possible generalizations for more than two arguments are also proposed.