Even though we w ill m ainly speak about semigroups, the concepts involved in this section are o f a universal algebraic nature and we refer to [5] and [7].
Definition 2,1.1.
A variety o f semigroups [monoids], in the sense o f B irkhoff, is a classo f semigroups [monoids] closed under taking subsemigroups [submonoids], quotients ÿ and direct products.
Proof. Since Ï Z C < K > m , it is clear that « K > m > s 3 < ^ > s.
Conversely, let V be a variety o f semigroups such that V D K. We w ill show that V D < K. > M .
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denoted by B [B M ], which we treat in the next section.
The intersection o f varieties is again a variety. Hence, since a ll semigroups
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[monoids] form a variety, given a class JC o f semigroups [monoids] there is a smallest | variety containing K, It is called the variety of semigroups [monoids] generated by K.N otation 2.1.2. Let A3 be a class o f monoids. We denote by < ^ >s and <
JC >u
the variety o f semigroups and the variety o f monoids generated by K, respectively. There are several possible descriptions o f the variety o f semigroups [monoids] generated by a certain class o f semigroups [m onoids]. One o f them is the follow ing statement, which is a specification, for semigroups [monoids] o f a general theorem on Universal Algebra, due to Tarski:
P roposition 2.1.3. Let K = {Si}i^i be a class o f semigroups. Then < K > s= { T € S : r < f J S '; - ,7 C I} ..
The follow ing result w ill be useful.
P roposition 2.1.4. Let JC be a class of monoids. Then « JC> M > S = < JC >s
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I f T 6 < K >M» then T < n»ei M ), where the Si (i £ I) are monoids o f AC. Since the Si (i € I) are in V and T also divides Yli^i Si in S, we deduce that T e y . Hence < AC > M Ç y .
We now present the notion o f free object in a class o f semigroups.
D efin itio n 2.1.5, Let AC be a class o f semigroups and A a set. Let (S', / ) be an A-
generated semigroup such that S' E AC.
I f fo r every T £JC and for every map a : A - ^ T there is a morphism à : S T
such that the follow ing diagram commutes
S a
T
a
We give next a construction o f the free object generated by a set A in the variety o f a ll semigroups.
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we say that S has the universal mapping property fo r AC over A, The set / ( A) is called a set o f free generators o f S' in AC and S is said to be the free object in AC generated by
A .
I f it exists, we w ill denote by FaHC) the free object in AC generated by A.
Rem ark 2.1.6. We notice that the morphism à o f 2.1.5 is unique. Moreover, à is
suijective if and only if a ( A ) generates T. |
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letters. A word in A is a non-empty finite sequence, w ritten x\...Xn> o f elements o f A . On the set o f a ll words define a binary operation, called concatenation by
( J/l • • • Vn) ~ • • • Vn-
This is an associative operation and A is a set o f generators o f .
Proposition 2.1.8. The semigroup A^ has the universal mapping property fo r S over
A .
We call A"*" the^ee semigroup on A.
The free monoid on A , denoted by A *, can be described in sim ilar manner. The only m odification required is the inclusion o f the empty word, denoted by 1. Thus A * = A + U {1 }.
Proposition 2.1.8 remains true w ith ‘‘semigroup” replaced by “ monoid" and w ith
A^ replaced by A *.
In the next propositions we present various useful properties o f A^ and A *. (See [18].)
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P roposition 2.1.9. Every word u E A^ has a unique factorization as a product o f
elements of A. !
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D efinition 2.1.10. A monoid M is called equidivisible if for every a,b,c,d £ M ,ab = cd im plies either o = cu, u6 = d fo r some u E A f, or ov = c, 6 = vd fo r some V E M .
Proposition 2.1.11. The free monoid A* is equidivisible.
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his result.
D efinition 2.1.12. Let A be a non-empty set and let 5 be a semigroup. Let it>, w' £ .
We say that the identity (or relation) w = is satisfied in S if p{w) = (p{w^) for every moiphism p : A^ S,
Notice that we can regard an identity w = (w , E A ^) on a semigroup S as a subset o f A^ x A^.
If AC is a class o f semigroups, we say that AC satisfies an identity w - w' iî each member o f AC satisfies tu = tu'. Moreover, if Z is a set o f identities, we say that AC
satisfies Z if every member o f AC satisfies every identity o f Z .
B irkho ff showed that given a variety V o f semigroups, there is a set Z o f identities such that V is precisely the class o f semigroups satisfying Z , that is, y is an equational class. We put y = [Z ].
N otation 2.1.13. If y is a variety, we denote by V the dual variety o f V, that is, the variety satisfying the dual identities.
Given a class AC = {y },ç /o fv a rie tie s o f semigroups such that V< = [ f t ] ( i E I) , then QAC = [IJ ie i f t ] ' However, the union o f varieties need not be a variety. One does have a w ell defined jo in o f a class AC = {Vf}<6i o f varieties: the intersection o f a ll varieties containing every Vi, denoted by V»€j A can be d ifficu lt to determine a “ good” set o f relations fo r the jo in o f a class o f varieties. We can however obtain a set o f defining identities in the follow ing way. (See [16].)
D efin itio n 2.1.14. A congruence 0 on a semigroup S is fully invariant if fo r every
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morphism tp: S -* S
The set o f fu lly invariant congruences on a semigroup S is closed under taking arbitrary intersections. Given a set o f relations R Ç x we denote by z/(R) the smallest fu lly invariant congruence on containing R. It is the fully invariant congruence generated by R.
If 7 = [ i2 ], sometimes we put V) for R) .
Proposition 2.1.15. Let K = {V i},g i be a class of varieties such that Vi - [Ri] ( i £
I) . Then
v ^ =
iei iei
The next statement yields that in any variety o f semigroups there are free objects. Proposition 2.1.16. Let R Ç A"*^ x A^ be a set of relations and let V be a variety of semigroups such that V = [B ]. Then A^ jv { R ) € V a n d A f j v i K ) is the free object in V generated by A,
Remark 2.1.17. Let 7 be a variety o f semigroups and le t [}v be the standard map embedding A in 7 ). Then, according w ith 2.1.5, fo r each semigroup S E V and each map a : A —* S v/e have the follow ing commutative diagram
a
Fa(V)
2 2
that is, there is a unique morphism 0 : 7 ) —> 5* such that = a;. Moreover, ^ is suijective if and only if a (A ) generates S,
The rest o f this section amounts to some sim plifying notation.
N otation 2.1.18. If S and T are semigroups, p is an equivalence relation on T and / : S' T is a map, let p / be the equivalence relation on S defined by
X p f v if fi x ) p f ( y ) . Sometimes we s till denote by p the relation pf.
N otation 2.1.19. If 7 is a variety o f semigroups, we denote by Cy ["R-y, C7y] the Green relation £ [H, J ] on Fa( 7 ). According to 2,1.18, we can extend this notation
w ithout confusion in order to interpret £ y as a relation on : if u, v E A*'^ then by u jCy t; we shall mean l}y(« ) Cy t)y(u ).
Rem ark 2.1.20. Let 7 and W be varieties o f semigroups such that 7 Ç P7. Then by 2.1.17 there is a unique (suijective) morphism dw,y • Fa(W ) —> Fa{ V ) such that the follow ing diagram commutes.
A* ÜÏ ► Fa(W )
Fa(V )
Moreover, the follow ing statements are equivalent (i) i C y ) y = iCw )y>
(ii) i C y ) y y = Cw-