Bisymmetric and quasitrivial operations: characterizations and enumerations

We investigate the class of bisymmetric and quasitrivial binary operations on a given set $X$ and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set $X$ is finite.Comment: arXiv admin note: text overlap with arXiv:1709.0916

On idempotent n-ary semigroups

This thesis, which consists of two parts, focuses on characterizations and descriptions of classes of idempotent n-ary semigroups where n >= 2 is an integer. Part I is devoted to the study of various classes of idempotent semigroups and their link with certain concepts stemming from social choice theory. In Part II, we provide constructive descriptions of various classes of idempotent n-ary semigroups. More precisely, after recalling and studying the concepts of single-peakedness and rectangular semigroups in Chapters 1 and 2, respectively, in Chapter 3 we provide characterizations of the classes of idempotent semigroups and totally ordered idempotent semigroups, in which the latter two concepts play a central role. Then in Chapter 4 we particularize the latter characterizations to the classes of quasitrivial semigroups and totally ordered quasitrivial semigroups. We then generalize these results to the class of quasitrivial n-ary semigroups in Chapter 5. Chapter 6 is devoted to characterizations of several classes of idempotent n-ary semigroups satisfying quasitriviality on certain subsets of the domain. Finally, Chapter 7 focuses on characterizations of the class of symmetric idempotent n-ary semigroups. Throughout this thesis, we also provide several enumeration results which led to new integer sequences that are now recorded in The On-Line Encyclopedia of Integer Sequences (OEIS). For instance, one of these enumeration results led to a new definition of the Catalan numbers

On the structure of symmetric nn-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 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

A new invariance identity and means

The invariance identity involving three operations $D_{f,g}:X\times X\rightarrow X$ of the form \begin{equation*} D_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \oplus g\left( y\right) \right) \text{,} \end{equation*} is proposed. The connections of these operations with means is investigated. The question when the invariance equality admits three means leads to a composite functional equation. Problem to determine its continuous solutions is posed

On bisymmetric and quasitrivial operations

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