COMPOSITIO MATHEMATICA R. GORTON A-systems Compositio Mathematica, tome 33, no 1 (1976), p. 3-13 <http://www.numdam.org/item?id=CM_1976__33_1_3_0> © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 33, Fasc. 1, 1976, pag. 3-13 Noordhoff International Publishing Printed in the Netherlands A-SYSTEMS* R. Gorton 1. Introduction The axiomatic study of the substitutive algebra of functions has its roots in the works of Schonfinkel [16], Curry [2] and Menger [8]. In 1959, Menger [12] introduced a set of axioms designed to describe the algebra of ordinary functions under addition, multiplication or composition. During the 1960’s this work was continued, notably, by Schweizer and Sklar [ 17, 18, 20, 2 1 ]. Their initial paper [17] discusses a set of five axioms which, together, are equivalent to the six axioms given by Menger [12]. Their later articles focus attention on the axiomatic study of composition. The algebra of functions III culminates in two representation theorems, one of which gives sufficient conditions for a function to be represented as a union of minimal functions called atoms [20]. The purpose of this paper is to axiomatically describe the substitutive or additive behavior of atoms. 2. Preliminaries An a-system is an ordered triple (A, °, ’) such that : Al. (A, o, ’) is an inverse semigroup with null element p. A2. If a, b ~ A and Ø ~ a 03BF b b 03BF b’ = a’ 03BF a. EXAMPLE (2.1): Let A consist of all restrictions of the identity function on the set S(|S| ~ 2). If "o" represents composition and, for any f E S, f = f ’, then (A, 0,’) is an inverse semigroup (with null element Ø) violating axiom A2. * Some of thèse results appeared in a thesis written by the author under the guidance of Prof. A. Sklar. 3 4 EXAMPLE (2.2): Let S be any non-empty set and let A = S x S ~ {Ø}. Define (a, b) 0 (c, d) = (c, b) if a = d ; otherwise (a, b) 0 (c, d) = 0. Then (A, o, ’) is an a-system where (a, b)’ = (b, a). EXAMPLE (2.3): Let (G, +) be any group. Let A = G x G u {Ø}, where (a, b) o (c, d) = (a, b + d) if a = c; otherwise (a, b) 0 (c, d) = 0. Also, for any (a, b) E A, Ø03BF (a, b) = 0 = (a, b)03BFØ. Then (A, o, ’) is an a-system where (a, b)’ = (a, - b). EXAMPLE (2.4): Let (R, + ,·) be any division ring. Let where 0: R - R is given by 0(x) = 0 for all x ~ R. Define f 03BF g by: (f 0 g)(x) = f(x) · g(x). Then (A, o, ’) is an a-system where f ’(x) = (f(x))-1 and 0=0. In the sequel, (A, o, ’) denotes an a-system. LEMMA (2.5): If a E A, then the following are equivalent: The other identity is proved similarly. LEMMA (2.7) : Let a(~ Ø), b(~ Ø) ~ A. Then either a 0 b 0 b’ = 0 or a 0 b 0 b’ = a. The latter case occurs if and only if a 0 b =/= 0. Dually, we have PROOF : Suppose a 0 b = 5 LEMMA (2.10) : If a, b E A and 0 =1= b = a 0 b then a = b 0 b’. Dually, if Ø + b ~ b 03BF a then a = b’03BFb. PROOF: 0 + b = a 0 b implies a’ 0 a = b 0 b’ whence (from Lemma 2.7) 0 + b 0 b’ = a 0 b 0 b’ = a. For any a ~ A, let La = a 03BF d, Ra = a03BF a. THEOREM (2.11) : (A,o,L,R) is a function system; i.e., (A,o,L,R) satisfies : PROOF : See [21; theorem 23]. 3. Categorical semigroups and Brandt semigroups If a, b are elements of any function system then a ~ b means a = b 0 Ra[21]. THEOREM (3.1): In any a-system, "~" is trivial; i.e., a, b E A, a ç b implies a = b or a = 0. PROOF : If a ~ b then a = boa’ 0 a. If a =1= Ø then, by Lemma 2.7, a = b. COROLLARY (3.2) : (A, o, L, R) is a categorical semigroup; i.e., (A, 0, L, R) possesses a zero element 0 satisfying Ro = Ø and 1. (A, 0) is a semigroup. 2. For all elements a E A, (a) LRa = Ra, RLa = La; (b) La 0 a = a = a 0 Ra. 3. For all a, b in A, a 0 b =1= Ø if and only if a =1= Ø, b ~ Ø and Ra = Lb[21]. PROOF : See [21; theorem 25]. 6 EXAMPLE (3.3): Let For any f(~ 0) E C, let L f, Rf E C be defined by : Lf(x) = Rf(x) = 0 for all x E 9t. Then (C, +, L, R) is a categorical semigroup violating axiom Al. LEMMA (3.4): Let (C, 0, L, R) be a categorical semigroup. If, for any a E C there exists x E C such that x 0 a = Ra, then a 0 x = La. PROOF : Let x 0 a = Ra =1= 0. Then Rx = La and whence a 0 x =1= Ø. Hence Ra = Lx. Now, there exists y E C such that y 0 x = Rx. Thus Ry = Lx and Ly = Rx. Hence y 0 x = La which implies that y 0 x 0 a = La 0 a = a. Thus y 0 Ra = y o Ry = y = a. L,EMMA (3.5) : If (C, 0, L, R) is a categorical semigroup such that for any a E C there exists a’ E C such that a’ 0 a = Ra then C is cancellative; i.e., and PROOF : Ø ~ b03BFa = c03BFa implies Rb = La = Rc whence by Lemma 3.4, b = b 0 Rb = b o La = boa 0 a’ = c03BFa03BFa’ = c03BFLa = coRe = c. THEOREM (3.6): Let (C,o,L,R) be a categorical semigroup. Then C is an a-system if and only if for any a E C there exists a’ E C such that a’ 0 a = Ra. PROOF : Suppose that C is a categorical semigroup having the above property. Then, for any a E C, a’ 0 a c a’ 0 a = a’ 0 a 0 Ra = a’ 0 a. By cancelling, we get: and Notice that a’ must be unique (by the cancellative property). Thus (C, 0,’) is an inverse semigroup with null element 0. Let a, b E C. If Ø ~ a03BFb then Ra = Lb; i.e., a’03BFa=b03BFb’. Conversely, let (C, 0, L, R) be a categorical semigroup which is also 7 an a-system. Let a(~ Ø) E C. Then Ø =1= a = a 0 Ra and the result follows from Lemma 2.10. THEOREM (3.7) : Let (A, 0, ’) be an a-system. If, for any two idempotents a, b E A there exists x E A such that a 0 x 0 b =1= Ø then A is a Brandt semi- group and conversely, where a Brandt semigroup i,s defined to be a semigroup (B, .) with zero element Ø satisfying: 1. If a,b,cEBand,if ac = bc ~ Ø or ca = cb ~~~~~~~~~~~~~~~~~~~~~~~ 2. If a, b, c E B and if ab 0 and bc =1= 0 then abc =1= 0. 3. For each a(~ Ø) in B there exists a unique e E B such that ea = a, a unique f E B such that af = a and a unique a’ E B such that a’a = f. 4. If e, f are non-zero idempotents of B, then there exists a E B such that eaJ =1= 0[l]. PROOF : Let (A, 0, ’) be an a-system. Let a(~ Ø) E A. Let e = a 0 a’ and f = a’ 0 a. Then e 0 a = a = a 0 f. Moreover, x 0 a = f implies x = a’ since A is cancellative. Conversely, let (B,o) be a Brandt semigroup. Then B is an inverse semigroup with null elements [1 ; page 102]. Let a 0 b. Then there exists e E B such that e 0 a 0 b = a 03BF b. Thus Hence COROLLARY (3.8): Every Brandt semigroup is a categorical semigroup. Then (A, o, ’) is an a-system which is not a Brandt semigroup. 4. Functions over A Let a, b be distinct elements of (A, o, ’). Then a and b are inconsistent if a’ 0 a = b’ o b (cf. [11 ; page 169]). 8 If a and b are distinct elements of A and a - Rb = b 0 Ra then either Ø = a 03BF Rb or a = a 03BF Rb. In the former case a’ 03BF a ~ b’ o b and in the latter case a = b. In either case, a and b are consistent. Conversely, let a and b be distinct consistent elements. Then a’ 03BF a ~ b’ 0 b. Hence a03BFb’03BFb = a o Rb = Ø. Similarly, b o Ra = Ø. Thus we have proved: LEMMA (4.1): Let a, b ~ A. Then a and b are consistent if and only if a 0 Rb = b 0 Ra[20].