Concerning syntax and notation, everything is now available to work with. We take this opportunity to have a closer look at the algebraic laws of relation algebra. In particular, we will be interested in how they can be traced back to a small subset of rules which can serve as axioms. We present them now and discuss them immediately afterwards.

We should stress that we work with heterogeneous relations. This contrasts greatly with the traditional work of the relation algebra community which is almost completely restricted to a homogeneous environment – possibly enhanced by cylindric algebra considerations. Some of the constructs which follow simply do not exist in a homogeneous context, for example the direct power and the membership relation. At first glance, it seems simpler to study homogeneous as opposed to heterogeneous relations. But attempting domain constructions in the homogeneous setting immediately leads necessarily to non-finite models. Also deeper problems, such as the fact that ╥A,B; ╥B,C = ╥A,C does not necessarily hold, have only recently come to attention; this applies also to unsharpness.

Laws of relation algebra

The set of axioms for an abstract (possibly heterogeneous) relation algebra is nowadays generally agreed upon, and it is rather short. When we use the concept of a category, this does not mean that we are introducing a higher concept. Rather, it is used here as a mathematically acceptable way to prevent multiplying a 7 × 5-matrix with a 4 × 6-matrix.

After one's first work with orderings, one will certainly come across a situation in which the concept of an ordering cannot be applied in its initial form, a high jump competition, for example. Here certain heights are given and athletes achieve them or not. Often, more than one athlete will reach the same maximum height, for example a whole class of athletes will jump 2.35 m high. Such a situation can no longer be studied using an order – one will switch to a preorder. We develop the traditional hierarchy of orderings (linear strictorder, weakorder, semiorder, intervalorder) in a coherent and proof-economic way. Intervalorders are treated in more detail since they have attracted much attention in their relationship with interval graphs, transitive orientability, and the consecutive 1s property. Block-transitive strictorders are investigated as a new and algebraically promising concept. Then we study how an ordering of some type can – by slightly extending it – be embedded into some more restrictive type; for example, a semiorder into its weakorder closure.

In a very general way, equivalences are related to preorders, and these in turn are related to measurement theory as used in physics, psychology, economic sciences, and other fields. Scientists have contributed to measurement theory in order to give a firm basis to social sciences or behavioral sciences. The degree to which a science is considered an already developed one depends to a great extent on the ability to measure.

It is shown that the syntactic algebra of the characteristic series of a rational language L is semisimple in the following two cases: L is a free submonoid generated by a bifix code, or L is a cyclic language.

This chapter has two appendices, one on semisimple algebras (without proofs) and another on simple semigroups, with concise proofs. We use the symbols A1 and A2 to refer to them.

Bifix codes

Let E be a set of endomorphisms of a finite dimensional vector space V. Recall that E is called irreducible if there is no subspace of V other than 0 and V itself which is invariant under all endomorphisms in E. Similarly, we say that E is completely reducible if V is a direct sum V = V1 ⊕…⊕ Vk of subspaces such that for each i, the set of induced endomorphisms e∣Vi of Vi, for e ϵ E, is irreducible.

A set of matrices in Kn×n (K being a field) is irreducible (resp. completely reducible) if it is so, viewed as a set of endomorphisms acting at the right on K1×n, or equivalently at the left on Kn×1 (for this equivalence, see Exercises 1.1 and 1.2).

A linear representation (λ, μ, γ)ofaseries S ϵ K⟪A⟫ is irreducible (resp. completely reducible) if the set of matrices {μa ∣ a ϵ A}(or equivalently the sets μA* or μ(K⟨A⟩)) is so.

This chapter gives a presentation of results concerning the minimization of linear representations of recognizable series. A central concept of this study is the notion of syntactic algebra, which is introduced in Section 2.1. Rational series are characterized by the fact that their syntactic algebras are finite dimensional (Theorem 1.2). The syntactic right ideal leads to the notion of rank and of Hankel matrix; the quotient by this ideal is the analogue for series of the minimal automaton for languages.

Section 2.2 is devoted to the detailed study of minimal linear representations. The relations between representations and syntactic algebra are given. Two minimal representations are always similar (Theorem 2.4), and an explicit form of the minimal representation is given (Corollary 2.3).

The minimization algorithm is presented in Section 2.3. We start with a study of prefix sets. The main tool is a description of bases of right ideals of the ring of noncommutative polynomials (Theorem 3.2).

Several important consequences are given. Among them are Cohn's result on the freeness of right ideals, the Schreier formula for right ideals and linear recurrence relations for the coefficients of a rational series. A detailed description of the minimization algorithm completes the chapter.

Syntactic ideals

We start by assuming that K is a commutative ring. The algebra of polynomials K ⟨A⟩ is a free K-module having as a basis the free monoid A*. Consequently, the set K ⟪A⟫ of formal series can be identified with the dual of K⟨A⟩.

This chapter describes the relations between rational series and languages.

We start with Kleene's theorem, presented as a consequence of Schützenberger's theorem. Then we describe the cases where the support of a rational series is a rational language. The most important result states that if a series has finite image, then its support is a rational language (Theorem 2.10).

The family of languages which are supports of rational series have closure properties given in Section 3.4. The iteration theorem for rational series is proved in Section 3.5. The last section is concerned with an extremal property of supports which forces their rationality; to prove it, we use a remarkable characterization of rational languages due to Ehrenfeucht, Parikh and Rozenberg.

Kleene's theorem

Definition A congruence in a monoid is an equivalence relation which is compatible with the operation in the monoid. A language L is recognizable if there exists a congruence with finite index in A* that saturates L (that is L is a union of equivalence classes).

It is equivalent to say that L is recognizable if there exists a finite monoid M, a morphism of monoids ϕ: A* → M and a subset P of M such that L = ϕ−1(P).

The product of two languages L1 and L2 is the language L1L2 ={xy∣x ϵ L1, y ϵ L2}. If L is a language, the submonoid generated by L is ∪n≥0Ln.