Multi-algebras from the viewpoint of algebraic logic

• Veröffentlicht in: Algebra and Discrete Mathematics
• Datum: 2003
• 1. Verfasser: Cırulis, J.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут прикладної математики і механіки НАН України 2003
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/154670
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Multi-algebras from the viewpoint of algebraic logic / J. Cırulis // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ.

Where U is a structure for a first-order language
 L
 ≈ with equality ≈, a standard construction associates with every
 formula f of L
 ≈ the set kfk of those assignments which fulfill f in
 U. These sets make up a (cylindric like) set algebra Cs(U) that
 is a homomorphic image of the algebra of formulas. If L
 ≈ does
 not have predicate symbols distinct from ≈, i.e. U is an ordinary
 algebra, then Cs(U) is generated by its elements ks ≈ tk; thus, the
 function (s, t) 7→ ks ≈ tk comprises all information on Cs(U).
 In the paper, we consider the analogues of such functions for
 multi-algebras. Instead of ≈, the relation ε of singular inclusion
 is accepted as the basic one (sεt is read as ‘s has a single value,
 which is also a value of t’). Then every multi-algebra U can be
 completely restored from the function (s, t) 7→ ks ε tk. The class
 of such functions is given an axiomatic description.