4 research outputs found
On the poset of computation rules for nonassociative calculus
The symmetric maximum, denoted by v, is an extension of the usual max
operation so that 0 is the neutral element, and -x is the symmetric (or
inverse) of x, i.e., x v(-x)=0. However, such an extension does not preserve
the associativity of max. This fact asks for systematic ways of parenthesing
(or bracketing) terms of a sequence (with more than two arguments) when using
such an extended maximum. We refer to such systematic (predefined) ways of
parenthesing as computation rules. As it turns out there are infinitely many
computation rules each of which corresponding to a systematic way of bracketing
arguments of sequences. Essentially, computation rules reduce to deleting terms
of sequences based on the condition x v(-x)=0. This observation gives raise to
a quasi-order on the set of such computation rules: say that rule 1 is below
rule 2 if for all sequences of numbers, rule 1 deletes more terms in the
sequence than rule 2. In this paper we present a study of this quasi-ordering
of computation rules. In particular, we show that the induced poset of all
equivalence classes of computation rules is uncountably infinite, has
infinitely many maximal elements, has infinitely many atoms, and it embeds the
powerset of natural numbers ordered by inclusion
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..