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Monoids (I = 1, 2) We Can Define � 1  C 2000 M Geometria Superiore Reference Cards Push out (Sommes amalgam´ees). Given a diagram i : P ! Qi Monoids (i = 1; 2) we can define 1 c 2000 M. Cailotto, Permissions on last. v0.0 u −!2 Send comments and corrections to [email protected] Q1 ⊕P Q2 := coker P −! Q1 ⊕ Q2 u1 The category of Monoids. 1 and it is a standard fact that the natural arrows ji : Qi ! A monoid (M; ·; 1) (commutative with unit) is a set M with a Q1 ⊕P Q2 define a cocartesian square: u1 composition law · : M × M ! M and an element 1 2 M such P −−−! Q1 that: the composition is associative: (a · b) · c = a · (b · c) (for ? ? all a; b; c 2 M), commutative: a · b = b · a (for all a; b 2 M), u2 y y j1 and 1 is a neutral element for the composition: 1 · a = a (for Q2 −! Q1 ⊕P Q2 : all a 2 M). j2 ' ' More explicitly we have Q1 ⊕P Q2 = (Q1 ⊕ Q2)=R with R the A morphism (M; ·; 1M ) −!(N; ·; 1N ) of monoids is a map M ! N smallest equivalence relation stable under product (in Q1 ⊕P of sets commuting with · and 1: '(ab) = '(a)'(b) for all Q2) and making (u1(p); 1) ∼R (1; u2(p)) for all p 2 P . a; b 2 M and '(1 ) = 1 . Thus we have defined the cat- M N In general it is not easy to understand the relation involved in egory Mon of monoids. the description of Q1 ⊕P Q2, but in the case in which one of f1g is a monoid, initial and final object of the category Mon. the Qi is a group we have a nice description: Q1 ⊕P Q2 = (Q ⊕ Q )=∼ where (x ; x ) ∼ (y ; y ) if and only if: (N; +; 0) and (Z; ·; 1) are monoids. 1 2 1 2 1 2 ( Free monoids. For any set S there is a free monoid with x1u1(d) = y1u1(c) S there exist c; d 2 P such that generators the elements of S: the free monoid on S: N . The S S x u (c) = y u (d) natural morphism S ! N given by s 7! es (element of N 2 2 2 2 sending s to 1 and the other elements to zero) realizes a (bi- i.e. if and only if there exist c; d 2 P such that (x1; x2)(u1(d); u2(c)) = ∼ S functorial) bijection HomSet(S; M) = HomMon(N ;M) for (y1; y2)(u1(c); u2(d)) in Q1 ⊕ Q2. We see in fact that ∼ is an any monoid M. This construction gives a left adjoint to the equivalent relation (without hypothesis) and, by the given hy- forgetful functor from Mon to Set. pothesis, it is just the relation described in the previous num- Associated Group. There exists a left adjoint to the natural ber. u forgetful functor from Ab to Mon, given by M 7! Mgp where: Cokernel. Given a morphism P ! Q of monoids, we can con- gp u 1 M := M ×M ∼ and ∼ is the equivalence relation on M ×M sider the diagram P −! Q −{1g and put coker(u) = coker(u; 1) := defined by (x; y) ∼ (m; n) if and only if there exists a 2 M coker(u; 1) that is: Q ⊕ f1g=∼ = Q=∼ where such that a · x · n = a · y · m. We have the natural morphism 1 1 M −! Mgp, x 7! (x; 1). x ∼ y if and only if there exist c; d 2 P such that xu(c) = yu(d) Projective limits. In the category Mon all projective limits (note that f1g is an abelian group!). exist, and the projective limit functor commutes with the forget It is useful to observe that if we have a cocartesian diagram functor from Mon to Set. In fact this functor is a right ad- u1 P −! Q1 joint, and so commutes with all projective limits, and we have ? ? a canonical monoid structure (component by component) on u2 y y j1 the (set theoretical) projective limits of a system of monoids. Q2 −! Q In particular we have (arbitrary) products and fiber products j2 of monoids. the induced morphism between the cokernel: coker(u1) ! coker(j2) is always an isomorphism. f Warning: Kernel. Given a morphism M ! N of monoids, we Warning: Cokernel. The notion of epimorphism (i.e. right have also the kernel kerf which is a particular projective limit; erasable map) in Mon is weaker than the notion of epimor- being the kernel of the double map f and the constant 1 map, −1 phism in Set (i.e. surjectivity): moreover an epimorphism has it is realized as for abelian groups by kerf = f (1). But a trivial cokernel, while a map with a trivial cokernel is not the notion of monomorphism (i.e. left eraseable map) in Mon necessarily an epimorphism. coindices with the notion of monomorphism in Set (i.e. injec- tivity): so a monomorphism has a trivial kernel, while a map Consider for example N := (N; +; 0) and the map N×N ! N×N with a trivial kernel is not necessarily a monomorphism (con- defined by (x; y) 7! (x; x + y); the map is not surjective, but s sider for example N := (N; +; 0) and the map N×N ! N defined its cokernel is trivial. Composing with N × N !(N; ·; 1) defined by the sum (x; y) 7! x + y). by s(e1) = 1 and s(e2) = 0, i.e. sending (x; y) to 1 if y = 0 Inductive limits. In the category Mon all inductive limits and 0 if y 6= 0, we find the same map as composing with the exist, but in general it is difficult to calculate them. map r defined by r(e1) = 0 and r(e2) = 0, i.e. sending all (x; y) 6= (0; 0) to 0. Direct sums. Given a family fMig the direct sum of the i2IL Q We can see however that in the category of integral monoids family is the submonoid of the product Mi ⊆ Mi i2I i2I the notion of epimorphism coincides with that of \map with of I-ple (s ) where s = 1 for almost all components. i i Mi trivial cokernel" while the notion of surjectivity is stronger. Cokernel of double arrows. We consider a double arrow u; v : Example: the canonical morphism P ! P gp is epi, because P Q. The cokernel of u and v is the quotient coker(u; v) := ⇒ P gp=P = 0 (remark however that the morphism is epi also in Q=R were R is the minimal equivalent relation stable by prod- n o the category Mon). uct and containing u(P ) × v(P ) := (u(p); v(p)) : p 2 P . P Quotients. In particular, if we have a submonoid Q of a monoid The canonical morphism Q ! Q=R has the attended universal P , we can define the quotient Q=P := coker(Q,! P ) = Q=∼ property. where x ∼ y if and only if there exist c; d 2 P such that xc = yd. 1 GeoSupRC - Monoids c 2000 by MC Presentation of monoids by generators and relations. If f : P ! Q is a morphism of finitely generated saturated We note that there is a bijection monoids, K a prime of Q with ht(K) = 1 and J = f (K) (I) ∼= (I) with ht(J) = 1, then the ramification index of f at K is the HomMon ;M −! M ' 7−! '(ei) gp N i2I integer n ≥ 0 such that vK ◦ f = nvJ . (I) where (ei)i2I is the canonical base of N : ei(j) = δi;j. Submonoids and Faces. We say that a subset R of a Monoid P is a submonoid if 1 2 R and one of the following, equivalent, A family (xi)i2I of elements of a monoid M generates M if (I) conditions is satisfied: the morphism N ! M sending ei 7! xi is surjective. (s ) R · R ⊆ R This means that every element x of M can be written as a 0 product x = xn1 xn2 ··· xn` with i 2 I and n 2 . (s) for all x 2 R and y 2 R one has xy 2 R i1 i2 i` j j N (I) (I) We say that Q ⊆ P is a saturated subset if the converse of Given two sets I and J, and elements rj 2 N ×N for every j 2 J, we can define the monoid M with generators (x ) and condition (s) holds, i.e. if xy 2 Q implies that x and y belong i i2I to Q. relations (rj)j2J as the cokernel of the double arrow: p1 (J) r (I) (I) −−−! (I) A face of P is a submonoid and a saturated subset, i.e. is a N −−−! N × N −−−! N subset F of P such that 1 2 F and xy 2 F if and only if x 2 F p 2 and y 2 F . ej 7−−−! rj We see immediately that taking the complement gives a bijec- and we will write the \relations" rj also as \equalities" p1(rj) = tion between the set of ideals (resp. submonoids) and the set p2(rj). of saturated subsets (resp. prime subsets) of a monoid. The Algebra of monoids over a ring. Let R be a commutative same is true for the set of prime ideals and the set of faces: a ring with unit. The forgetful functor from the category of R- face is the complement of a prime ideal. algebra to Mon given by (A; +; ·) 7! (A; ·) admits a left adjoint M 7! R[M] where R[M] is the free R-module with bases ex for Let P be a monoid: x 2 M and multiplication defined by exey = ex·y and R- (i) If an ideal contains 1 it is the monoid P .
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