20 research outputs found
Reducibility of n-ary semigroups: from quasitriviality towards idempotency
Let be a nonempty set. Denote by the class of associative operations satisfying the condition whenever at least of the elements are equal to each other. The elements of are said to be quasitrivial and those of are said to be idempotent. We show that and we give conditions on the set for the last inclusions to be strict. The class was recently characterized by Couceiro and Devillet \cite{CouDev}, who showed that its elements are reducible to binary associative operations. However, some elements of are not reducible. In this paper, we characterize the class and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides
Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)
International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..
On the structure of symmetric -ary bands
We study the class of symmetric -ary bands. These are -ary semigroups
such that is invariant under the action of permutations and
idempotent, i.e., satisfies for all . We first
provide a structure theorem for these symmetric -ary bands that extends the
classical (strong) semilattice decomposition of certain classes of bands. We
introduce the concept of strong -ary semilattice of -ary semigroups and
we show that the symmetric -ary bands are exactly the strong -ary
semilattices of -ary extensions of Abelian groups whose exponents divide
. Finally, we use the structure theorem to obtain necessary and sufficient
conditions for a symmetric -ary band to be reducible to a semigroup
Decomposition schemes for symmetric n-ary bands
We extend the classical (strong) semilattice decomposition scheme of certain classes of semigroups to the class of idempotent symmetric n-ary semigroups (i.e. symmetric n-ary bands) where n \geq 2 is an integer. More precisely, we show that these semigroups are exactly the strong n-ary semilattices of n-ary extensions of Abelian groups whose exponents divide n-1. We then use this main result to obtain necessary and sufficient conditions for a symmetric n-ary band to be reducible to a semigroup
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
Characterizations of quasitrivial symmetric nondecreasing associative operations
We provide a description of the class of n-ary operations on an arbitrary
chain that are quasitrivial, symmetric, nondecreasing, and associative. We also
prove that associativity can be replaced with bisymmetry in the definition of
this class. Finally we investigate the special situation where the chain is
finite
Reducibity of n-ary semigroups: from quasitriviality towards idempotency
Let be a nonempty set. Denote by the class of associative operations satisfying the condition whenever at least of the elements are equal to each other. The elements of are said to be quasitrivial and those of are said to be idempotent. We show that . The class was recently characterized by Couceiro and Devillet, who showed that its elements are reducible to binary associative operations. However, some elements of are not reducible. In this paper, we characterize the class and show that its elements are reducible. In particular, we show that each of these elements is an extension of an -ary Abelian group operation whose exponent divides
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
Visual characterization of associative quasitrivial nondecreasing functions on finite chains
In this paper we provide visual characterization of associative quasitrivial nondecreasing operations on finite chains. We also provide a characterization of bisymmetric quasitrivial nondecreasing binary operations on finite chains. Finally, we estimate the number of functions belonging to the previous classes