Lattice Concepts of Module Theory

Confluence and termination of fuzzy relations

“…A complete residuated lattice L is called a chain if it is linearly ordered. If for any a i 2 L (i 2 I) there is a finite subset I 0 # I such that W i2I a i ¼ W i2I 0 a i then L is called a Noetherian [7] (complete) residuated lattice. All properties of complete residuated lattices used in the sequel are well known and can be found, e.g., in [2,9,12,13].…”

Cofinitely Supplemented Modular Lattices

“…A lattice L is said to be compact if 1 is compact and compactly generated (or algebraic) if each of its elements is a join of compact elements [6]. For compactly generated compact lattices a supplement of an element is compact [3,Proposition 12.2 (2)]. In the following proposition we show that for an arbitrary lattice L a supplement of a cofinite element is compact: …”

“…We show in We give proofs of the results for lattices when the proofs are different from those in the module case. All definitions and related properties not given here can be found in [3,4].…”

Continuous fuzzy Horn logic

“…(ii) This is a note for readers familiar with [5,6]. The characterization of quasivarieties presented in [6] is restricted to residuated lattices whose lattice part is a so-called complete Noetherian lattice [7]. Such a subclass of structures of truth degrees includes e. g. all finite residuated lattices but does not include residuated lattices on [0, 1] given by left-continuous t-norms.…”