Generalizations of lattices via non-deterministic operators

• Published in: Discrete mathematics Vol. 295; no. 1; pp. 107 - 141
• Main Authors: Martínez, J., Gutiérrez, G., de Guzmán, I.P., Cordero, P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 28.05.2005; Elsevier
• Subjects: Algebra; Combinatorics. Ordered structures; Exact sciences and technology; General algebraic systems; Lattice theory; Mathematical analysis; Mathematics; Multilattice; Multisemilattice; Non-deterministic operator; Operator theory; Order, lattices, ordered algebraic structures; Sciences and techniques of general use; Universal multilattice; Universal multisemilattice; Multilattice; Universal multilattice; Lattice theory; Universal multisemilattice; Non-deterministic operator; Multisemilattice; Algebraic structure; Ordered structure; Operator theory; Lattice; Lattice theory: Non-deterministic operator; Finite element method; Upper bound; Partially ordered set; End; Lattice structure
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2004.08.043

Benado (Čehoslovak. Mat. Ž. 79(4) (1954) 105–129) and later Hansen (Discrete Math. 33(1) (1981) 99–101) have offered an algebraic characterization of multilattice (i.e., a poset where every pair of elements satisfies that any upper bound is greater than or equal to a minimal upper bound, and also satisfies the dual property). To that end, they introduce two algebraic operators that are a generalization of the operators ∧ and ∨ in a lattice. However, in Martinez et al. (Math. Comput. Sci. Eng. (2001) 238–248), we give the only algebraic characterization of the multisemilattice structure that exists in the literature. Moreover, this characterization allows us to give a more adequate characterization of the multilattice structure. The main advantage of our algebraic characterizations is that they are natural generalizations of the semilattice and lattice structures. It is well-known that in the lattice theory we can use indistinctly pairs of elements or finite subsets to characterize them. However, this is not true when we work with multilattices. For this reason in this paper we introduce two new structures from the ordered point of view, called universal multisemilattice and universal multilattice, and we propose an equivalent algebraic characterization for them. These new structures are generalizations, on one hand, of semilattice and lattice and, on the other hand, of multisemilattice and multilattice, respectively. The algebraic characterizations have the same advantages as the two introduced by us in Martinez et al. The most important purpose of this paper is to deepen the theoretical study of universal multisemilattices and universal multilattices.