Algebras of relations of various ranks, some current trends and applications

Here the emphasis is on the main pillars of Tarskian structuralist approach to logic: relation algebras, cylindric algebras, polyadic algebras, and Boolean algebras with operators. We also tried to highlight the recent renaissance of these areas and their fusion with new trends related to logic, like the guarded fragment or dynamic logic. Tarskian algebraic logic is far too broad and too fruitful and prolific by now to be covered in a short paper like this. Therefore the overview part of the paper is rather incomplete, we had to omit important directions as well as important results. Hopefully, this incompleteness will be alleviated by the accompanying paper of Tarek Sayed Ahmed [81]. The structuralist approach to a branch of learning aims for separating out the really essential things in the phenomena being studied abstracting from the accidental wrappings or details (called in computer science “syntactic sugar”). As a result of this, eventually one associates to the original phenomena (or systems) being studied streamlined elegant mathematical structures. These streamlined structures can be algebras in the sense of universal algebra, or other kinds of elegant well understood mathematical structures like e.g. space-time geometries in the case of relativity theory. In the present case they will be algebras, but as we said, this is not essential to the approach, they could be geometric metric spaces, topologies, etc. Along the above structuralist lines, one arrives at a kind of duality theory between the theory describing the original systems (or phenomena) and the corresponding streamlined mathematical structures. The important point in calling this a duality is the requirement that from the streamlined abstract mathematical structures one should be able to reconstruct the originals up to a certain equivalence determined by what we consider essential in our first step mentioned above. So far, two expressions showed up which we will use in the above outlined sense. These are “structuralist approach to something”, and “duality theory” (in this structuralist spirit). Algebraic logic can be regarded as a structuralist approach to the subject matter of logic. In other words, algebraic logic sets up a duality theory between the world of logics and the world of classes of algebras. ⋆ Research supported by Hungarian National Foundation for Scientific Research grant No T43242 and by COST grant No 274. The algebraic counterpart of classical propositional logic is the world of Boolean algebras (BA’s for short). BA theory was immensely successful in helping and improving, clarifying classical propositional logic, see e.g. [32]. What happens then if we want to extend the algebraization of classical propositional logic yielding BA’s to first-order logic? Well, BA’s are algebras of unary relations. That is, the elements of a BA B are unary relations and the operations of B are the natural operations on unary relations e.g. intersection, complementation. The problem of extending this approach to predicate logics boils down to the problem of expanding the natural algebras of unary relations to natural algebras of relations of higher ranks, i.e. of relations in general, more precisely, algebras of not necessarily binary relations e.g. algebras of ternary relations. The reason for this is, roughly speaking, the fact that the basic building blocks of predicate logics are predicates, and the meanings of predicates can be relations of arbitrary ranks. Indeed, already in the middle of the last century, when De Morgan wanted to generalize algebras of propositional logic in the direction of what we would call today predicate logic, he turned to algebras of binary relations. That was probably the beginning of the quest for algebras of relations in general. Returning to this quest, the new algebras will, of course, have more operations than BA’s, since between relations in general there are more kinds of connections than between unary relations (e.g. one relation might be the converse, sometimes called inverse, of the other). The framework for the quest for the natural algebras of relations is universal algebra. The reason for this is that universal algebra is the field which investigates classes of algebras in general, their interconnections, their fundamental properties. Therefore universal algebra can provide us for our search with a “map and a compass” to orient ourselves. There is a further good reason for using universal algebra. Namely, universal algebra is not only a unifying framework, but it also contains powerful theories. E.g. if we know in advance some general properties of the kinds of algebras we are going to investigate, then universal algebra can reward us with a powerful machinery for doing these investigations. Among the special classes of algebras concerning which universal algebra has powerful theories are the so called discriminator varieties and the arithmetical varieties . At the same time, algebras originating from logic turn out to fall in one of these two categories, in most cases. Let us return to our task of moving from BA’s of unary relations to expanded BA’s of relations in general. What are the elements of a BA? They are sets of 1 De Morgan illustrated the need for expanding the algebras of unary relations (i.e. BA’s) to algebras of relations in general (the topic of the present paper) by saying that the scholastics, after two millennia of Aristotelian tradition, were still unable to prove that if a horse is an animal, then a horse’s tail is an animal’s tail. (“v0 is a tail of v1” is a binary relation.) “points”. What will be the elements of the expanded new algebras? One thing about them seems to be certain, they will be sets of sequences. Why? Because relations in general are sets of sequences. These sequences may be just pairs if the relation is binary, they may be triples if the relation is ternary, or they may be longer — or more general kinds of sequences. (There is another consideration pointing in the direction of sequences. Namely, the semantics of quantifier logics is defined via satisfaction of formulas in models, which in turn is defined via evaluations of variables, and these evaluations are sequences. The meaning of a formula in a model is the set of those sequences which satisfy the formula in that model. Thus we arrive again at sets of sequences.) So, one thing is clear at this point, namely that the elements of our expanded BA’s of relations will be sets of sequences. Indeed, this applies to practically all known algebraizations of predicate logics or quantifier logics. Summing up: the key paradigm for moving from BA’s to richer versions of algebraic logic is replacing sets of points with sets of sequences as the elements of our algebras; e.g. in the case of relation algebras, our algebras are complex algebras of oriented graphs i.e. their elements are sets of “arrows”. This is where the name “arrow logic” comes from, cf. [16], [62]. In the more general case of longer sequences, our algebras are like complex algebras of lists in the sense of the programming language LISP, so we can regard a sequence or a list as a generalization of an arrow, say like a longer arrow which has points in the middle, too. 1 Algebras of binary relations Let us start concentrating on the simplest nontrivial case, namely that of the algebras of binary relations. Actually, these algebras will be strong enough to be called a truly first-order (as opposed to propositional) algebraic logic, namely [29], [87, §5.3] show that the logic captured by binary relation algebras is strong enough to serve as a vehicle for set theory and hence for ordinary metamathematics. A binary relation algebra (BRA for short) is an algebra whose elements are binary relations and whose operations are the operations of taking the union R∪S of two relations R,S, taking the complement −R of R, taking the relation algebraic composition R ◦ S = {〈a, b〉 : ∃c(aRc and cSb)} of two relations R,S 2 When explaining the so-called “holy grail of modern physics” (grand unification theories) in his video “The Elegant Universe”, Brian Greene said that the key idea was replacing point-like building blocks by string-like blocks (actually: superstrings), in their paradigm-shift. The motivation he gave was that strings have more “degrees of freedom” than points do (strings can wriggle while points cannot). The same can be said about switching from points to sequences which we are describing here (sequences can wriggle, too). So perhaps the two paradigm shifts (the one described here and the one in cosmology) might turn out to have something in common. and taking the inverse R = {〈b, a〉 : 〈a, b〉 ∈ R} of the binary relation R. That is, Definition 1. A BRA is an algebra A = 〈A;∪,−, ◦, 〉 where (i) A is a nonempty set of binary relations, (ii) 〈A;∪,−〉 is a Boolean set algebra, i.e. R,S ∈ A ⇒ R ∪ S,−R ∈ A, and (iii) A is closed under taking relation algebraic composition and inverses, i.e. R,S ∈ A ⇒ R ◦ S,R ∈ A. BRA denotes the class of all algebras isomorphic to BRA’s. ⊓⊔ Item (ii) in the definition of a BRA above implies thatA has a biggest element, called the unit of A. Let V = ⋃ A be this unit. Then (iii) above implies that V ◦ V ⊆ V and V −1 ⊆ V , hence V is an equivalence relation. Thus the unit of a BRA is always an equivalence relation. Let us return to universal algebra as a unifying framework. If A is a BRA as above, then the similarity type or signature of A consists of the function symbols ∪,−, ◦, −1 where the first two are Boolean join and complementation. Homomorphisms, equations etc. are defined accordingly; e.g. homomorphisms should preserve all four operations, and (x∪ y) ◦ z = (x ◦ z)∪ (y ◦ z), (x∪ y) = x ∪ y are typical equations. We will often denote by 0 and 1 the Boolean zero and unit, i.e. the smallest and biggest elements of the Boolean algebra. Notice that a BRA is a BA enriched with further operations ◦, , and these extra-Boolean operations all distribute over ∪. Such algebras are called Boolean algebras with operators (BAO’s for short). A BAO 〈B;∪,−, fi : i ∈ I〉 is called normal if the Boolean zero 0 is a zero-element for all the extra-Boolean operations fi, i.e. if fi(x1, . . . , xn) = 0 whenever one of the arguments x1, . . . , xn is 0. The literature of BAO’s is quite extensive, see e.g. [46], [47] to mention two papers. Having a fresh look at our BRA’s with an abstract algebraic eye, we notice that they should be very familiar from the abstract algebraic literature. Namely, a BRA A = 〈A;∪,−, ◦, 〉 consists of two well known algebraic structures, (R1) a Boolean algebra 〈A;∪,−〉 and 3 In the literature various other symbols are used for these operations, perhaps the most often used symbols are +,−, ; ,` . We hope it will not lead to confusion that we use the same symbols for the operations and the operation symbols denoting these operations. 4 These two equations express that composition and inverse distribute over ∪. (R2) an involuted semigroup 〈A; ◦, 〉 sharing the same universe. Further, the connection between these two structures is described by saying that (R3) A = 〈A;∪,−, ◦, 〉 is a normal BAO. These properties are all expressible with equations. The above (R1)-(R3) define a nice equational class V containing BRA and is a reasonable starting point for an axiomatic study of the algebras of relations. Postulates like above already appear in De Morgan [23], and since then investigations of relation algebras have been carried on for almost 130 years. The question naturally arises whether V equals BRA. It will turn out that BRA indeed can be defined by a set E of equations, but this set E of equations cannot be chosen to be finite. Thus V is bigger than BRA since V is defined by a finite set of equations. To illuminate the use of universal algebraic methods in algebraic logic, first we recall some standard material from Universal Algebra. Subdirect products of algebras, and subdirectly irreducible algebras are defined in practically every textbook on Universal Algebra. An algebra A is subdirectly irreducible if it is not (isomorphic to) a subdirect product of algebras all different from A or the one-element algebra. A subdirect product is a subalgebra of a product of algebras, having a special “economy” property. For our purposes, this “economy” property will not be needed. By Birkhoff’s classical theorem, every algebra is a subdirect product of some subdirectly irreducible ones. Therefore, the subdirectly irreducible algebras are often regarded as the basic building blocks of all the other algebras. In particular, when studying an algebra A, it is often enough to study its subdirectly irreducible building blocks. For example, the unique (up to isomorphism) subdirectly irreducible BA is the 2-element one. It is not hard to check that the subdirectly irreducible BRA’s are exactly the isomorphic copies of the BRA’s whose unit element as an equivalence relation has exactly one equivalence block, i.e. whose unit element is a Descarte space U × U . Definition 2. A class K of algebras is called a discriminator class iff conditions (i),(ii) below hold. (i) There is a term τ(x, y, z, u) in the language of K such that in every subdirectly irreducible member of K we have