COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
250 Takayuki Tamura Dejinition. Let G(0) and G’(P) be groupoids with binary operations 8, 8’ respectively. If there are three bijections h, q, r of G(8) to G’(0’) such that (xOy)r = (xh)@(yq) for all x, y E G (e), then we say that G(8) is isotopic to G’(V). If it is necessary to specify h, q, r, we say G(8) is (h, q, r)-isotopic to G’(0’). We denote it by G(B) 25 (h. 4. I) G’(W) or G(8) z G’(0’). THEOREM 2.2. Let S and T be a fixed set and a semigroup respectively. Let (a,~)o = fx,,, (a, PI@’ = eg,, a, fl E T. SZ@T is restrictedly isomorphic onto SY’o T with respect to the projections of SZ eT and SXelTto T tf and only tf there is an automorphism a +a’ of T and a system (fa; a E T) of permutations of S such that a groupoid S(e,,,) is (f,, fp, f,)-isotopic to S(OiP,P) for all a,BET. Let 9 and d be relations on a semigroup D. As usual the product e .CY of e and o is defined by p.0 = {(x,Y); (4 4 E 0, (z, y) E u for some z E D}. Let u) = DxD, L = {(x, x); x E D}. THEOREM 2.3. A semigroup D is gp-decomposable if and only if there is a congruence q on D and an equivalence o on D such that in which (2.6) can be replaced by e-0 = co, eno=L, (2.6) (2.7) a*,o = co. (2.6’) Then D % (D/o)K(D/e) where D/e is the factor semigroup of D module q and D/a is the factor set of D module o. We know many examples of general products: Direct product, semidirect product [3], [6], group extension [3], Rees’ regular representation of completely simple semigroups [l], the representation of commutative archimedean cancellative semigroups without idempotent [ll], Q-semigroups [15], and so on. 2.4. Left general product. As a special case of a general product, we make the Definition. A general product SFeT is called a left general product of S by T if and only if (a, B)@ = (c(, y)@ for all a, B, y E T. (2.8) SYeT is called a right general product of S by T if and only if (a, /?)@ = (y, p)@ for all a, /3, y E T. (2.8’) Semigroups and groupoids 251 In case (2.8), 8, B depends on only a, so 8,, B is denoted by e,.. Then (2.4) is rewritten : 8 OT. a* empe= 0,. h ea. for all a, ? E T, all a C S. (2.9) In case (2.8’), 8,, B is independent of CC, and 8,> B is denoted by 8., and (2.4) is e .(L (I * e., = ttcls +k II e., for all a, fi E T, all a E S. (2.9’) A left congruence is a left compatible equivalence, namely an equivalence 0 satisfying xoy * zxazy for all z. THEOREM 2.4. Let D be a semigroup. D is isomorphic onto a left general product of a set S by a semigroup T if and only if there is a congruence e on D and a left congruence o on D such that and D/q = T, 1 D/u ! = j S 1 o-0 = w (equivalently 0.q = o), ena = L. EXAMPLE. Let T be a semigroup, F a set, and let x denote a mapping of F into T: lx = aa where ;Z E F, aA E T. The set of all mappings x of F into T is denoted by S. For /? E T and x E S we define an element /3*x as follows: lx = aA * &9*x) = PaA. Then uh4*x = B*(Y-x). A binary operation is defined on G = SX T as follows: (x, a)(v, B> = (a*% 40. (2.10) Then G is a semigroup with respect to (2.10) and it is a left general product of S by T. Further the semigroup G with (2.10) is completely determined by a semigroup T and a cardinal number m = IF[, and G is denoted by G = G5%(T). We can describe the structure of B&+K) in terms of the semigroup of this kind. THEOREM 2.5. Let m = /El - 1 and Se be the full transformation semigroup over E (cf 111). %3&S) is isomorphic onto CBJ,,(%~). 2.5. Sub-generalproduct. In Q 2.3 we found that the two concepts, h-homomorphism and general product, are equivalent. What relationship does there exist between general products and homomorphisms?
252 Takayuki Tamura Let U be a subset of SFT, and define Prj,(U) = {a E T; (~3 a> E U}. Dejinition. If U is a subsemigroup of Sy@T and if p*j,( U) = T, then U is called a sub-general product of SXoT. In the following theorem, the latter statement makes the theorem have sense. THEOREM 2.6. If a semigroup D is homomorphic onto a semigroup T under a mapping g, then D is restrictedly isomorphic into SY@T with respect to g and the projection of SEoT to Tfor some S. Furthermore there exists an SO among the above S such that 1 S o j is either the minimum of 1 S/ or possibly the minimum plus one. Proof. Let D = U D,, D,g = a, Clearly 1 D,] e 1 D j for all a E T. The set =ET {lDol[; a E T} has a least upper bound. (For this the well-ordered principle is used.) Let m = 1 f1.u.b. {/DE 1; a E T} and take a system of sets S, of symbols such that 1 S, 1 = m for all a E T and a set So with I So I = m. Further we assume that D, & S, and S, contains a special symbol 0,, 0, t Dw and So contains a special symbol 0. Now let fb be a bijection of S to S, such that Of, = 0,. We define a binary operation on G = SX T as follows: (x, a)(y, j) = ~~~$$xG1~ aI@ ~~f2ra.s-$EDp i Then we can prove that G = SFoT where K~f,)(Yfs)>f$ XQ9Y = xf=EDg, YfpED, { 0 otherwise. Let D’ = {(x, a); xf, E D,, a E T). Then Prj, (D’) = T and D’ 2 D under b, a> -, xf,, a E T. 2.6. Construction of some general products. As a simplest interesting example of general product, we will construct all general left products of a set S by a right zero semigroup T. ra B Let T = {a, B), a a j B a B The equations (2.9) are Semigroups and groupoids e cc. cl* es. = 8,. %a 6,. 6. a% 8,. = e,. +kc, 0,. 8 L1. 0 j+ 8,. = 8,. +ka 0,. %I+ (I++ e,. = eB. +kc, 0,. 253 (2.11) 8,. and %,. are semigroup operations. In order to construct all left general products G = S??,T we may find all ordered pairs of semigroup operations on S: (%a., RJ.> which corresponds to G = G&L) U GK’$.>, ( G, 1 = 1 GB /. For fixed T and S, G is denoted by G(%,., %,.). Clearly G(%,., %,.) z G(%,., %a.). Instead of ordered pairs it is sufficient to find pairs (%,., %@,) regardless of order. Let G,s denote the set of all semigroup operations defined on S. (Gs contains isomorphic ones.) We define a relation w on Gs as follows: %-~ifandonlyif%a~~=%~Aca~and~.~%=~~cn%forallaES. The relation N is reflexive symmetric. Let YE, y: be transformations of S defined by zcp: = zex, zv: = X~Z respectively. Then 8 =+++~cy = %+#, 7 for all a E S, if and only if y&p; = @y$ for all x, y E S. As special cases we will determine the relation N on Gs in the case /SI=Z3. I. Left generalproduct of S, ISI = 2, by right zero semigroup T. a b Let S = {a, b}. m , x, y, z, u = a or b, is the table a x y b,z ru Explanation of the notations which will be used later: For example 4 denotes the semigroup abl b a j , i.e. 40 = 4. 4i denotes lbai i abi which is the isomorphic image of 4 under 1
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250 Takayuki Tamura<br />
Dejinition. Let G(0) and G’(P) be groupoids with binary operations 8, 8’<br />
respectively. If there are three bijections h, q, r of G(8) to G’(0’) such that<br />
(xOy)r = (xh)@(yq) for all x, y E G (e),<br />
then we say that G(8) is isotopic to G’(V). If it is necessary to specify h, q,<br />
r, we say G(8) is (h, q, r)-isotopic to G’(0’). We denote it by<br />
G(B) 25<br />
(h. 4. I)<br />
G’(W) or G(8) z G’(0’).<br />
THEOREM 2.2. Let S and T be a fixed set and a semigroup respectively. Let<br />
(a,~)o = fx,,, (a, PI@’ = eg,, a, fl E T. SZ@T is restrictedly isomorphic onto<br />
SY’o T with respect to the projections of SZ eT and SXelTto T tf and only tf<br />
there is an automorphism a +a’ of T and a system (fa; a E T) of permutations<br />
of S such that a groupoid S(e,,,) is (f,, fp, f,)-isotopic to S(OiP,P) for all<br />
a,BET.<br />
Let 9 and d be relations on a semigroup D. As usual the product e .CY<br />
of e and o is defined by<br />
p.0 = {(x,Y); (4 4 E 0, (z, y) E u for some z E D}.<br />
Let u) = DxD, L = {(x, x); x E D}.<br />
THEOREM 2.3. A semigroup D is gp-decomposable if and only if there is a<br />
congruence q on D and an equivalence o on D such that<br />
in which (2.6) can be replaced by<br />
e-0 = co,<br />
eno=L,<br />
(2.6)<br />
(2.7)<br />
a*,o = co. (2.6’)<br />
Then D % (D/o)K(D/e) where D/e is the factor semigroup of D module q and<br />
D/a is the factor set of D module o.<br />
We know many examples of general products: Direct product, semidirect<br />
product [3], [6], group extension [3], Rees’ regular representation of<br />
completely simple semigroups [l], the representation of commutative<br />
archimedean cancellative semigroups without idempotent [ll], Q-semigroups<br />
[15], and so on.<br />
2.4. Left general product. As a special case of a general product, we<br />
make the<br />
Definition. A general product SFeT is called a left general product of<br />
S by T if and only if<br />
(a, B)@ = (c(, y)@ for all a, B, y E T. (2.8)<br />
SYeT is called a right general product of S by T if and only if<br />
(a, /?)@ = (y, p)@ for all a, /3, y E T. (2.8’)<br />
Semigroups and groupoids 251<br />
In case (2.8), 8, B depends on only a, so 8,, B is denoted by e,.. Then (2.4) is<br />
rewritten :<br />
8 OT. a* empe= 0,. h ea. for all a, ? E T, all a C S. (2.9)<br />
In case (2.8’), 8,, B is independent of CC, and 8,> B is denoted by 8., and (2.4)<br />
is<br />
e .(L (I * e., = ttcls +k II e., for all a, fi E T, all a E S. (2.9’)<br />
A left congruence is a left compatible equivalence, namely an equivalence<br />
0 satisfying<br />
xoy * zxazy for all z.<br />
THEOREM 2.4. Let D be a semigroup. D is isomorphic onto a left general<br />
product of a set S by a semigroup T if and only if there is a congruence e on D<br />
and a left congruence o on D such that<br />
and<br />
D/q = T, 1 D/u ! = j S 1<br />
o-0 = w (equivalently 0.q = o),<br />
ena = L.<br />
EXAMPLE. Let T be a semigroup, F a set, and let x denote a mapping of<br />
F into T:<br />
lx = aa where ;Z E F, aA E T.<br />
The set of all mappings x of F into T is denoted by S. For /? E T and x E S we<br />
define an element /3*x as follows:<br />
lx = aA * &9*x) = PaA.<br />
Then<br />
uh4*x = B*(Y-x).<br />
A binary operation is defined on G = SX T as follows:<br />
(x, a)(v, B> = (a*% 40.<br />
(2.10)<br />
Then G is a semigroup with respect to (2.10) and it is a left general product<br />
of S by T. Further the semigroup G with (2.10) is completely determined by<br />
a semigroup T and a cardinal number m = IF[, and G is denoted by<br />
G = G5%(T).<br />
We can describe the structure of B&+K) in terms of the semigroup of this<br />
kind.<br />
THEOREM 2.5. Let m = /El - 1 and Se be the full transformation semigroup<br />
over E (cf 111). %3&S) is isomorphic onto CBJ,,(%~).<br />
2.5. Sub-generalproduct. In Q 2.3 we found that the two concepts, h-homomorphism<br />
and general product, are equivalent. What relationship does there<br />
exist between general products and homomorphisms?