We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...

We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 ! n ! !. Completely analogous statement holds for the case n !. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCA n+k . We prove that the complementa...

New successes in dealing with set theories by means of state-of-the-art theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) Tarski-Givant map calculus. In this paper we carry out this task in detail, setting the ground for a number of experiments. Key words: Set theory, relation algebras, first-order theorem-proving, algebraic logic. 1 Introduction Like other mature fields of mathematics, Set Theory deserves sustained efforts that bring to light richer and richer decidable fragments of it [5], general inference rules for reasoning in it [23, 2], effective proof strategies based on its domain-knowledge, and so forth. Advances in this specialized area of automated reasoning tend, in spite of their steadiness, to be slow compared to the overall progress in the field. Many experiments with set theories have hence been carried out with standard theorem-proving systems. Still today such experiments pose consider...

. Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra) . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin-Monk-Tarski [1985]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic equality-free quasi-polyadic algebras with rectangular atoms. The er...

In this paper a strong relationship is demonstrated between fork algebras and quasi-projective relation algebras. With the help of Tarski's classical representation theorem for quasi-projective relation algebras, a short proof is given for the representation theorem of fork algebras. As a by-product, we will discuss the difference between relative and absolute representation theorems. Fork algebras, due to their expressive power and applicability in computing science, have been intensively studied in the last four years. Their literature is alive and productive. See e.g. Baum, Frias, Haeberer, Veloso [33], [34], [3], [7], [8], [6], [9], [10] and Sain--Simon [27]. As described in the textbook [15] 2.7.46, in algebra, there are two kinds of representation theorems: absolute and relative representation. For fork algebras absolute representation was proved almost impossible in [18], [27], [26] 1 . However, relative representation is still possible and is quite useful (see e.g. [2]). The...

This paper is a survey of recent results concerning connections between relation algebras (RA), cylindric algebras (CA) and polyadic equality algebras (PEA). We describe exactly which subsets of the standard axioms for RA are needed for axiomatizing RA over the RA-reducts of CA 3 's, and we do the same for the class SA of semi-associative relation algebras. We also characterize the class of RA-reducts of PEA 3 's. We investigate the interconnections between the RA-axioms within CA 3 in more detail, and show that only four implications hold between them (one of which was proved earlier by Monk). In the other direction, we introduce a natural CA-theoretic equation MGR + , a generalization of the well-known Merry-Go-Round equation MGR of CA-theory. We show that MGR + is equivalent to the RA-reduct being an SA, and that MGR + implies that the RA-reduct determines the algebra itself, while MGR is not su#cient for either of these to hold. Then we investigate how di#erent CA's a single ...

The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representation theorems. The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski. The categorical approach to relational calculus was initiated by Mac Lane and Puppe, and Dedekind categories were introduced by Olivier and Serrato. The representation problem for Boolean relation algebras was proposed by Tarski as the question whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations. There are many sufficient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the...