Relativized Relation Algebras
نویسندگان
چکیده
In this paper we introduce a finitely axiomatizable equational class of relation type algebras which we call REL. This class includes all relation algebras and is closed under relativization to an arbitrary element. Thus REL contains every relativization of ReU , where ReU is the relation algebra of all binary relations on the set U . The main theorem is the REL Representation Theorem 5.4, in which every complete atomic algebra in REL is shown to be isomorphic to a relativization of a representable relation algebra, and hence, since every algebra in REL has a perfect extension in REL, every algebra in REL is isomorphic to a subalgebra of a relativization of some ReU . This characterizes REL. The paper is organized as follows. In §1 we define the class REL by giving a finite equational axiomatization. We also prove some elementary facts about algebras in REL that are needed for later sections. In §2 we show that REL is closed under relativization to arbitrary elements. In §3 we study a construction for adjoining converses, and in §4 we study a construction for adjoining identity elements. In §5 we use the results of §3 and §4 to prove the REL Representation Theorem, making use of the WA Representation Theorem (Theorem 5.20 of Maddux [2]).
منابع مشابه
Undecidable Relativizations of Algebras of Relations
In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic. ?
متن کاملPreservation of Sahlqvist fixed point equations in completions of relativized fixed point Boolean algebras with operators
We define Sahlqvist fixed point equations and relativized fixed point Boolean algebras with operators (relativized fixed point BAOs). We show that every Sahlqvist fixed point equation is preserved under completions of conjugated relativized fixed point BAOs. This extends the result of Givant and Venema [8] to the setting of relativized fixed point BAOs.
متن کاملConnections between cylindric algebras and relation algebras
We investigate the class SRaCAn for 4 n < ! and survey some recent results. We see that RAn | the subalgebras of relation algebras with relational bases | is too weak, and that the class of relation algebras whose canonical extension has an n-dimensional cylindric basis is too strong to deene the class. We introduce the notion of an n-dimensional hyperbasis and show that for any relation algebr...
متن کاملOrdered Models of the Lambda Calculus
: Answering a question by Honsell and Plotkin, we show that there are two equations between λ-terms, the so-called subtractive equations, consistent with λ-calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of λ-calculus, by studying the connection between the noti...
متن کامل