Semigroups in Complete Lattices

Patrik Eklund develops applications based on many-valued representation of information. Information typically resides in the form of expressions and terms as integrated in knowledge structures, so that term functors, extendable to monads, become important instrumentations in applications. Categorical term constructions with applications to Goguen's category have been recently achieved (cf. Fuzzy Sets and Syst. 298, 128-157 (2016)). Information representation supported by such monads, and as constructed over monoidal closed categories, inherits many-valuedness in suitable ways also in implementations.Javier Gutiérrez García has been interested in many-valued structures since the late 1990s. Over recent years these investigations have led him to a deeper understanding of the theory of quantales as the basis for a coherent development of many-valued structures (cf. Fuzzy Sets and Syst. 313 43-60 (2017)).Since the late 1980s the research work of Ulrich Höhle has been motivated by a non-idempotent extension of topos theory.  A result of these activities is a non-commutative and non-idempotent theory of quantale sets which can be expressed as enriched category theory in a specific quantaloid (cf. Fuzzy Sets and Syst. 166, 1-43 (2011), Theory Appl. Categ. 25(13), 342-367 (2011)). These investigations have also led to a deeper understanding of the theory of quantales. Based on a new concept of prime elements, a characterization of semi-unital and spatial quantales by six-valued topological spaces has been achieved (cf. Order 32(3), 329-346 (2015)). This result has non-trivial applications to the general theory of C*-algebras.Since the beginning of the 1990s the research work of Jari Kortelainen has been directed towards preorders and topologies as mathematical bases of imprecise information representation. This approach leads to the use of category theory as a suitable metalanguage. Especially, in cooperation with Patrik Eklund, his studies focus on categorical term constructions over specific categories (cf. Fuzzy Sets and Syst. 256, 211-235 (2014)) leading to term constructions over cocomplete monoidal biclosed categories (cf. Fuzzy Sets and Syst. 298, 128-157 (2016)).

“This monograph is a detailed and extensive investigation of the theory of quantales in different areas of mathematics. … the book is self-contained, well-organized and well-written and is warmly recommended to read.” (Ali Madanshekaf, zbMATH 1491.06001, 2022)

• Introduction.- 1 Foundations.- 2 Fundamentals of Quantales.- 3 Module Theory in Sup.- Appendix.- References.- Index.