Projectivity in (bounded) integral residuated lattices

In this paper we study projective algebras in varieties of (bounded) commutative integral residuated lattices from an algebraic (as opposed to categorical) point of view. In particular we use a well-established construction in residuated lattices: the ordinal sum. Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property that every finitely presented algebra is projective. In particular, we obtain results on (Stonean) Heyting algebras, certain varieties of hoops, and product algebras. Moreover, we study varieties with a Boolean retraction term, showing for instance that in a variety with a Boolean retraction term all finite Boolean algebras are projective. Finally, we connect our results with the theory of Unification

Categories of Residuated Lattices

We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on extended Priestley duality in which the ternary relation dualizing a residuated multiplication may be viewed as the graph of a partial function. We also present a new Esakia-like duality for Sugihara monoids in the spirit of Dunn\u27s binary Kripke-style semantics for the relevance logic R-mingle

The Reticulation of a Universal Algebra

The reticulation of an algebra $A$ is a bounded distributive lattice ${\cal L}(A)$ whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of $A$, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra $A$ from a semi-degenerate congruence-modular variety ${\cal C}$ in the case when the commutator of $A$, applied to compact congruences of $A$, produces compact congruences, in particular when ${\cal C}$ has principal commutators; furthermore, it turns out that weaker conditions than the fact that $A$ belongs to a congruence-modular variety are sufficient for $A$ to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from ${\cal C}$ is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and ${\cal C}$, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.Comment: 29 page

Publications of the Jet Propulsion Laboratory July 1965 through July 1966

Bibliography on Jet Propulsion Laboratory technical reports and memorandums, space programs summary, astronautics information, and literature searche

Annual Report of the University, 1972-1973, Volumes 1-3

At the varsity level, our teams have competed in the following sports: football, basketball, track, cross country, baseball, tennis, wrestling, swimming, golf, gymnastics and skiing. Junior varsity teams played regular schedules in football and basketball. A total of 167 athletes received major letter awards; 21 freshmen athletes were awarded numerals in basketball and football making a grand total of 188