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Fig. 6.1. The eggbox of D, fora regular x 6.1 Regular Semigroups Regular D-classes

Figure 6 1. The eggbox of D, fora regular x 6.1 Regular Semigroups Regular D-classes

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(5) Let S = {a, b,c} be a set of three elements, and define the product in the table. Monoids and groups Let S (that is, (S,-)) be a semigroup. An element x € S is a left identity of S, if
(5) Let S = {a, b,c} be a set of three elements, and define the product in the table. Monoids and groups Let S (that is, (S,-)) be a semigroup. An element x € S is a left identity of S, if
Lemma 2.1. An equivalence relation p on S is a congruence if and only if for all
Lemma 2.1. An equivalence relation p on S is a congruence if and only if for all
xp-(yp- zp) = zp- (yz)p = (x(yz))o = ((ay)z)p = (zy) p- zp = (ap- yp) - zp.
xp-(yp- zp) = zp- (yz)p = (x(yz))o = ((ay)z)p = (zy) p- zp = (ap- yp) - zp.
Proof. The proof is quite similar to the previous one. Here we observe that the mapping G defined by 6(2p) = a(x) is well defined, since Theorem 2.4 (Isomorphism theorem). Let a: S — P be a homomorphism. Then First of all, @ is well defined, that is, it is a function, because
Proof. The proof is quite similar to the previous one. Here we observe that the mapping G defined by 6(2p) = a(x) is well defined, since Theorem 2.4 (Isomorphism theorem). Let a: S — P be a homomorphism. Then First of all, @ is well defined, that is, it is a function, because
Lemma 4.7. If two elements x and y in S have the same spine wo, w1,..., U2m, ther Proof. If
Lemma 4.7. If two elements x and y in S have the same spine wo, w1,..., U2m, ther Proof. If
Proof. First we have to prove that p, maps R, into R,. For this, let z € Rr, that is, wl —.. »@l Wha hava nau
Proof. First we have to prove that p, maps R, into R,. For this, let z € Rr, that is, wl —.. »@l Wha hava nau
Fig. 5.5. Combination of Green’s lemmas Lemma 5.6. Let e € Es be an idempotent. If rLe, then re = x. If rRe, then ex = zx. Lemma 5.5. Let S be a semigroup, yRz, and let r,r’ € S* be such that yr = z and zr’ = y. Then
Fig. 5.5. Combination of Green’s lemmas Lemma 5.6. Let e € Es be an idempotent. If rLe, then re = x. If rRe, then ex = zx. Lemma 5.5. Let S be a semigroup, yRz, and let r,r’ € S* be such that yr = z and zr’ = y. Then
Fig. 5.6. Location theorem Proof. Suppose first that zy € R, M Ly. Since yLay, we may choose s = « in Lemma 5.4, and so pz: Ry — Rzy is a bijection. Also zyRx, and hence Ry, = Rz, which means that p,: R, — R, is abijection. The mapping p, preserves £-classes and SO p; maps R, M Lz onto R, M Ly = Hz. Therefore there exists az € R, M Lz such that p,(z) = #, thatis, a7 = 2, ye R,NLy — > R,NL,z contains a unique idempotent. Theorem 5.2. Let x,y € S. Then
Fig. 5.6. Location theorem Proof. Suppose first that zy € R, M Ly. Since yLay, we may choose s = « in Lemma 5.4, and so pz: Ry — Rzy is a bijection. Also zyRx, and hence Ry, = Rz, which means that p,: R, — R, is abijection. The mapping p, preserves £-classes and SO p; maps R, M Lz onto R, M Ly = Hz. Therefore there exists az € R, M Lz such that p,(z) = #, thatis, a7 = 2, ye R,NLy — > R,NL,z contains a unique idempotent. Theorem 5.2. Let x,y € S. Then
Fig. 6.2. The figures drawn by the words w and v (2) Consider Z x Z as a drawing board so that you can draw a unit line up wu, down d, right r and left 2. A word w € A* = {u,d,r,@}* produces a figure on Z x Z, when you start from (0,0) and follow the instructions given by the word w. Example 6.2. (1) If 9 is a group, then it is an inverse semigroup, and x~' is the group inverse of the element x.
Fig. 6.2. The figures drawn by the words w and v (2) Consider Z x Z as a drawing board so that you can draw a unit line up wu, down d, right r and left 2. A word w € A* = {u,d,r,@}* produces a figure on Z x Z, when you start from (0,0) and follow the instructions given by the word w. Example 6.2. (1) If 9 is a group, then it is an inverse semigroup, and x~' is the group inverse of the element x.
Let £ be a class of semigroups. Let F € £ be a semigroup in this class, X a nonempty set, and »: X — F an injective mapping into F' such that y(X) generates F’. We say that the pair (Fw) is a free £-semigroup on X, if for any P € £ and any mapping ag: X —> P there exists a homomorphism a: F' — P such that the following dianqram commiuitec thatic nn — nip
Let £ be a class of semigroups. Let F € £ be a semigroup in this class, X a nonempty set, and »: X — F an injective mapping into F' such that y(X) generates F’. We say that the pair (Fw) is a free £-semigroup on X, if for any P € £ and any mapping ag: X —> P there exists a homomorphism a: F' — P such that the following dianqram commiuitec thatic nn — nip
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