prime decomposition theorem for finite state sequential machines. Further, by applying this theorem together with the powerful solvability criteria of. Feit and ...
ALGEBRAICTHEORY OF MACHINES.I. PRIME DECOMPOSITIONTHEOREM FOR FINITE SEMIGROUPSAND MACHINES BY
KENNETH KROHN1.1)AND JOHN RHODES
Introduction. In the following all semigroups are of finite order. One semigroup Si is said to divide another semigroup S2, written Si|S2, if Si is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Si, with connecting homomorphism Y, is written S2 Xy Si. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S\(S2Xy Si) implies S|S2 or S|Si. It is shown that S is irreducible if and only if either: (i) S is a nontrivial simple group, in which case S is called a prime; or (ii) S is one of the four divisors of a certain three element semigroup U3 (see Definition 2.1) in which case S is called a unit. We remark that an anti-isomorphism of a unit need not be a unit. Thus the theory is not symmetric. The explanation is that semidirect product can be written from the left or from the right. Let be a collection of finite semigroups. We define K(Sf) as the closure of Sunder the operations of division and semidirect product. See Def-
inition 3.2. Then it is proved that SG K{Sf U {U3\) if and only if
PRIMES (S) C PRIMES (SS). Here PRIMES (S) = {P\ P is a nontrivial simple group and P divides S\ and PRIMES (SS) =(J {PRIMES (S)\S G Sf ). In particular, Se#(PRIMES
(S) U {U3\). A counterexample to
the conjecture that SG 7/f(IRR(S)) justifies the distinction between primes and units as well as the inclusion of U3 in the above formulas. A novel feature of this paper is the use of functions on free semigroups, i.e. machines, to prove facts about finite semigroups. These above results are obtained as an immediate corollary of a more general theorem (proved here) which finds application as the basis for a prime decomposition theorem for finite state sequential machines. Further, by applying this theorem together with the powerful solvability criteria of Feit and Thompson and of Burnside, we find that Corollary 4.1 answers in important cases the question "What machines can be constructed by series-parallel from counters, delays and units?" See §4. A heuristic discussion of this paper occurs in [6]. Received by the editors July 16, 1963. (^This research was sponsored in part by the Office of Naval Research, Information Systems
Branch, Contract Number: Nonr-4138(00).
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algebraic theory of machines. i
451
Both authors want to thank Professor Warren Ambrose for his important encouragement in the early days of this work. 1. Elementary properties of machines. Notation 1.1. In this paper A, B, C, • • • will denote nonempty sets. ZA denotes the free noncommutative semigroup without identity on the generators A. A machine will be any mapping /:£A —>Bi2). The natural "exten-
sion" f: £A->£ß
is defined by f(au • •-,an) = (/(a,), • ••>/foi, •••,«•))* We
also write / as (/)'. Let h: A —>B. Then h is the unique extension of A to a homomorphism £A into£ß. Thus h(au-..,an) = (h(ai), ■■■,h(an)).
of
Definition 1.1. Let /: £ A —> B and g: £ C—>D. Then f\g, read / divides g, if and only if there exists a homomorphism H: ^ A —C
h: D—>Bso that/=
and a function
hgH.
Definition 1.2. Let /: £A->P. Then S/ is the semigroup given by the congruence =/ on zZ^- Here t = fr if and only if f(atß) = f(arß) for all a,ß in Z A or a,/3 empty. The equivalence class containing t will be denoted by [t\. The mapping jf. S/—>B sending [t]f to f(t) induces the partition P( on Sf and (S,, Py) is termed the normal form of /. NF(/) = (Sf,Pf). Definition 1.3. Let (Si, Pi) and (S2, P2) be two semigroups with parti-
tions. Then (Si, Px)| (S2,P2), read (Si.PO divides (S2,P2), if and only if there exists a subsemigroup SQS2 and a homomorphism of S onto Si so
that s = s' (modP2) implies #(s) = a>(s')(modPi). Si|S2 if and only if Si is a homomorphic image of a subsemigroup S c S2Definition 1.4. Let S be a semigroup. Then /s: £ (S) —>S, read the machine of S, is defined by /s(si, • ■->sn) = IT"=isi- Let ^
be a collection of
semigroups. Then fy ={fs\S G ^ }• Proposition 1.1. Lei /: £A —B S/by hf(a) = [a\. Then
(1.1)
f = jffs,h,
showing / divides jffSf. Further, we define the homomorphism Hf. —»ZA by Hf(s) = (oi, • • -,a„). Here (au ■■•,an) is any fixed sequence of ^ A
such that [(au---,an)]f= proving (a).
s. Then jffsf = fHf showing jffSf divides / and
We next show that f\g implies NF(/) |NF(g).
Suppose hgH = f and con-
sider S'g= j [PT(r)]gGSg\t G^ZA I- S« is a subsemigroup of Sg since BTis a homomorphism. Then [H(f)]«—>[t]/ is a well-defined homomorphism
of S'g
(2) See references [3], [4], [6], and [7], and §4 of this paper for a discussion of machines and automata. See reference [ 5] for group theory and references [ l] and [ 8] for semigroup theory.
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452
kenneth
krohn and john rhodes
[April
onto Sf satisfying the conditions of Definition 1.3. This last assertion is proved by direct verification. We now show that NF(/)|NF(g) implies f\g. Let S'g be a subsemigroup of Se and a homomorphism of Sg onto Sf satisfying the conditions of Definition 1.3. Now by (a) it is sufficient to show that jffsf\jgfsgLet j'g be jg restricted to S'e. And let H: ZS/—be a homomophism such that for each s£S/, H(s) = i~with D such that (2.1)
g = hn+lgnnngn_l---h2glhl
where each g, is a finite direct sum of members of 3* and each A, for i = 1,
■•■,n + 1 isa function. Hereg;: ^A^—>Ai2 for i = 1,• • •,and Aj: C—»An, A2:Au-*Ati, •••,*»: A„_i2->A„i and An+1:An2->D. Each A;for i = 1, is a length preserving homomorphism.
(b) We cannot infer /, G SP( 9) from /, | f2 and f2 G SP( 9).
For example,
it can be shown that L\ divides a member of SP(j/i/3j) (see equations (3.1)) but does not lie in SP({/u3 [). However, the theorem of this paper implies that SP(/v U \fu3,Di}) is closed under division.
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454
kenneth
krohn and john rhodes
[April
Definition 2.3. Let S have the property that for all Si,S2 and Y, S\S2 XySi implies S\S2 or S|Si. Then S is said to be irreducible. IRR denotes the set of all irreducible semigroups.
Theorem,
(i) Let f: J^A^> B be a machine with Sf of finite order. Then
/eSP(/vU{/Ji,/[/3|)
if and only if PRIMES (Sf)C PRIMES ($S). In
particular
(2.2)
/£ SP(/primes(S/)U {DufU3)).
(ii) PRIMES U UNITS = IRR. 3. Proof of the theorem. In this section we write F(A,B) for the set of all mappings of A into B. The proof proceeds via several lemmas. First we give a converse to Proposition 1.2.
Lemma 3.1. Let S, and S2 be semigroups. Then fS2vSl£ SP({ fSvf^, Dufa}). Proof. One shows by direct computation
that fs2ws1equals
ht((bh X • • • X (6)„ X d)h3(g X DSl X d)°h~2(g X fSl)%. Here Si = {«i, • • •, s„} and g = fRX •■■X fR (taken n times) and R = Rf\Si,s2)-
RX--XR (taken n times) is R". Further, b = fa and (6); = b. Also d = fRl with Ri = RSv Here kt: S2wSi->RnxSi and hi(l,s) = (I, ■■-,l,s). Further h2: R" X Si -»R" X Si X Si and h2(lu •■■,ln,s) = (lu
ther h3:RnX(Si\j{*\)XSi—>
s, s). Fur-
(S2)nX Si with
A3(/i, • • •, /„, s, s') = (li(sis), • • •, ln(sns), s')
and *=1.
Finally A4:S2"X Si->S2wSi
and M*i, ••',*»») - (*»*) with
Ks.)= h Now for any finite set A,RA is a subsemigroup of a suitably large finite direct sum of Ux with itself. Further a restriction of a suitably large finite direct sum of Dx with itself yields DA. Thus the above expression lies in
SP({fsvfs2,Di,fUl}) and Lemma 3.1 is proved.
Lemma3.2. PRIMES U UNITS C IRR. Proof. We first show that PRIMES c IRR. Let G' £ PRIMES and G'|S2XySi. Thus there exists a subsemigroup S QS2Xy Si and a homomorphism
