On the $n$-ary algebras, semigroups and their universal covers

Aczélian n-ary semigroups

We show that the real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczél (1949); there symmetry was redundant.

A description of n-ary semigroups polynomial-derived from integral domains

We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Głazek and Gleichgewicht's classification of the corresponding ternary semigroups.

Every quasitrivial n-ary semigroup is reducible to a semigroup

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.

On the structure of symmetric $n$-ary bands

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 $F(x,\ldots,x)=x$ 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 divi…

允许邻接中性元素的不可约多元半群

Irreducible Polyadic Semigroups Admitting the Adjunction of a Neutral Element

本文指出文献[4]关于n元半群可邻接中性元素当且仅当可约化为二元半群的断言在n为奇数时不成立，并完整刻画了奇数n下允许邻接中性元素的不可约n元半群类。

Representations of Menger $(2,n)$-semigroups by multiplace functions

Investigation of partial multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for $n$-place functions is an $(n+1)$-ary superposition $[ ]$, but there are some other naturally defined operations, which are also worth of consideration. In this paper we consider binary Mann's compositions $\op{1},...,\op{n}$ for partial $n$-place functions, which have many important applications for the study of binary and $n$-ary operations. We present methods of represen…