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The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...) |
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These notes are a corrected and revised version of notes which accompanied lectures given at the Banach Center in the fall of 1991. The intent is to give a self-contained introduction to cylindric algebras from the concrete point of view. I hope that after reading this introduction the reader will be able to digest the basic works on this subject results about them as we go along. And we will try to motivate the notions from logic. See the end of (...) |
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A new system of algebraic logic is described. it is closely related to cylindric and polyadic algebras, and is axiomatized by a small number of very simple equations. |
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Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA ω 's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable, as well). We will also show that the complexity of the equational theory of RPEA ω is also extremely high in the recursion theoretic (...) |
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Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) |
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The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic. |
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This paper is about pairing relation algebras as well as fork algebras and related subjects. In the 1991-92 fork algebra papers it was conjectured that fork algebras admit a strong representation theorem . Then, this conjecture was disproved in the following sense: a strong representation theorem for all abstract fork algebras was proved to be impossible in most set theories including the usual one as well as most non-well-founded set theories. Here we show that the above quoted conjecture can still (...) |
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Among others we will see that the equational theory of ω dimensional representable polyadic equality algebras. We will also see that the complexity of the equational theory of RPEAω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko Sain and Viktor Gyuris [10], the following methodological conclusions will be drawn: the negative properties of polyadic algebras can be removed by switching from what we call the `polyadic algebraic paradigm' (...) |
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We give an algebraic version of first order logic without equality in which the class of representable algebras forms a finitely based equational class. Further, the representables are defined in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 , while its logical form is Corollary 5.2.For first order logic with equality we give a result weaker than the one for first order logic without equality. Namely, in (...) |
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