Algebra of Kleene

In Mathematical, a algebra of Kleene (of the name of the American logician Stephen Sticks Kleene) corresponds to the one of the two following concepts:

a lattice ordered and distributive with a Involution satisfying the Lois of Morgan and the inequality x∧−x ≤ y∨−y. With the result that each Boolean algebra is an algebra of Kleene, the reciprocal one being false. Following the example Boolean algebra which are based on the traditional logical proposals, the algebras of Kleene are based on the ternary logic of Kleene.
an algebraic structure which generalizes the operations known starting from rational expressions. The continuation of this article will treat this concept of algebra of Kleene. At the origin of this concept one finds the mathematician John Horton Conway who introduced it under the name of regular algebras .

Definition

Many nonequivalent definitions of the algebras of Kleene and dependant structures were given in the literature. We will give here the definition which seems most commonly allowed today.

An algebra of Kleene is a Ensemble equipped with both laws of composition interns +: With × → has and · : With × → has and of the operator *: With → A. These laws and this operator are noted has + B, ab and a* respectively. These operations satisfy the Axiome S following:

Associativeness of + and · : + (B has + c) = (has + b) + C and has (bc) = (ab) C for all has, B, C of A.
Commutativité of +: + B has = B + has for all has, B of A.
Distributivité: (B has + c) = (ab) + (ac) and (B + C) has = (Ba) + (Ca) for all has, B, C of A.
neutral elements for + and · : there exists an element 0 of has such as for all has a: has + 0 = 0 + has = A. There exists an element 1 of has such as for all has a: a1 = 1a = A.
0 are a element absorbing: a0 = 0a = 0 for all has A.

The axioms above define a semi-ring. One adds moreover:

the Idempotence of +: for all has of has, has + has = A.

It is consequently possible to define a Préordre ≤ on has while postulating has ≤ B if and only if has + B = B (or in an equivalent way, ≤ B has if and only there exists C in has such as has + C = b). This relation of order makes it possible to pose the last two axioms on the operator *:

1 + has (a*) ≤ a* for all has A.
1 + (a*) has ≤ a* for all has A.
if has and B are two elements of then has such as ab ≤ B a*b ≤ B.
if has and B are two elements of then has such as Ba ≤ B B (a*) ≤ B.

In an intuitive way, one can think of has + B like the union or smallest raising of has and B, and with ab like an increasing multiplication, in the direction where ≤ B has implies ax ≤ bx. The subjacent idea with the operator “star” is that a*=1 + has + aa + aaa +… From the point of view of the theory of the programming, one can interpret + like an operator of “choice” not determinist, · like the “sequential composition” and * like the “iteration”.

Examples

Are Σ a unit finished (a “alphabet”) and With the whole of the rational expressions on Σ. Two rational expressions are equal if they describe same the rational Langage. The unit has form an algebra of Kleene. In fact, it is even about a free algebra of Kleene in the direction where any valid equation in this algebra is valid in any algebra of Kleene.

That is to say Σ an alphabet. Maybe With the whole of the rational languages on Σ (or the whole of the languages without contexts on Σ, or even the whole of all the recursive languages, or finally the whole of all the languages on Σ). The union (written +) and the Concatenation (written •) of two elements of has belong to has, and thus the operation of Fermeture of Kleene is a Endomorphisme of A. One then obtains an algebra of Kleene with like element 0, the Empty set and like element 1, the unit which contains only the null string. Either M a Monoid with like identity an element E and or With the whole of the Sous-ensemble S of Mr. Are S and T two subsets of M, or S + T the union of S and T and ST = {chain: S in S and T in T}. S* is defined like the under-monoid of M generated by S, intuitively it corresponds to {E} ∪ S ∪ S ∪ SSS ∪… has form then an algebra of Kleene with like 0, the empty set and like 1, the singleton {E}. One can make a similar construction in very small category.

Let us suppose M a unit and With the whole of the binary relations on Mr. One can provide M with the three operations of the algebras of Klenne, namely the union for +, the composition for • and reflexive and transitive closing for *. The two constants are for 0 the relation empties (which does not connect anything) and for 1 the full relation (which connects all). Thus equipped, (M, +, •, *; 0,1) is an algebra of Kleene.

All Boolean algebra provided with operations v and ^ proves to be an algebra of Kleene if one identifies v with +, ^ with •, that one postulates a* = 1 for all has and that the 0 for the algebra of Kleene are the 0 of the Boolean algebra and of the same for 1.

A specific algebra of Kleene is useful when it is a question of calculating shorter ways in the balanced directed graphs: maybe With the completed real Right , let us pose which + B has is the minimum of has and B and which ab is the sum of real has and of the real B (the sum of +∞ and - ∞ being by +∞ definition). a* is real number zero if has is positive or null and - ∞ if has is strictly negative. Has is an algebra of Kleene in which 0 are the value - ∞ and 1 is real number zero.

Properties

Zero, noted 0, are the smallest element of the unit, in other words 0 ≤ has for all has in A.

The sum has + B is the small one raising of has and b: one has ≤ has + B and B ≤ has + B and if X is an element of has with has ≤ X and B ≤ X, then has + B ≤ X. In a similar way, a1 +… + year is smallest raising elements a1,…, year.

The multiplication and the addition are monotonic: if ≤ B has, then + X ≤ B has + X, ax ≤ bx and xa ≤ xb for any X of A.

Considering the operation *, we have 0* = 1 and 1* = 1, this * is increasing (has ≤ B implies a* ≤ b*), and that aⁿ ≤ a* for entire naturalness N. Moreover, (a*) (a*) = a*, (a*) * = a*, and has ≤ b* if and only if a* ≤ b*.

The formal series form an algebra of Kleene on the condition of taking for F * the series (1 - F ) ^-1.

If has is an algebra of Kleene and N a natural entirety, one can consider the unit Mⁿ(A) made up of all matrices N by N with entries of A. By using the ordinary concepts of additions and matric multiplications , one can define a single operation * such as M_n(A) becomes an algebra of Kleene.

History

The algebras of Kleene were not defined by Kleene. It introduced the rational expressions and postulated the existence of a complete unit of axioms which would make it possible to derive all the valid equations in the rational expressions. The first axioms were proposed by John H. Conway and a system of complete axioms defining the algebras of Kleene and solving the problem was proposed by Dexter Kozen. who showed to them complétude.

See too

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