\magnification=1200

\input AMSSYM.DEF
\def\Q{\mathord{\Bbb Q}}
\def\R{\mathord{\Bbb R}}
\def\N{\mathord{{\Bbb N}}}
\def\Z{\mathord{{\Bbb Z}}}

\def\monoid{*monoid}
\def\condition{$(\bf po)$}
\def\po{\mathrel{\leq}}
\def\supp{\mathop{{\rm supp}}}
\def\tmo{\mathrel{\preceq}}
\def\stmo{\mathrel{\prec}}

\def\graver{\mathop{\rm Gr}}
\def\mb{\mathop{\rm MinBal}}



\noindent {\bf Lecture 1. Monoids: General algebraic aspects}

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\noindent {\bf Definition.} A monoid is a set equipped with and associative binary operation that has an identity element.

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\noindent Given a monoid $M$, we have the following data:
\smallskip
\item{i)} a set $|M|$;
\item{ii)} a map $\ast _M:|M|\times|M|\to |M|; (x, y)\mapsto x*_My$, satisfying $(x\ast _M y)\ast _M z =  x\ast _M (y\ast _M z)$ for all $x,y,z\in |M|$;
\item{iii)} an element $e_M\in |M|$, satisfying $x\ast _M e_M =  x = e_M \ast _M x$ for all $x\in |M|$.
 
\smallskip\noindent Often we take liberties with the notation, using the name $M$ to refer to the set $|M|$ or suppressing the subscript on $e$ or on $\ast $.

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\noindent {\bf Definition.} A {\it submonoid\/} of $M$ is a subset that: $i$) contains $e$, and $ii$) contains $x\ast y$ whenever it contains $x$ and $y$.  Given a subset $X\subseteq M$, the {\it submonoid generated by $X$\/}, denoted $\langle X\rangle_M$, is the intersection of all submonoids of $M$ that contain $X$.  We say $M$ is {\it finitely generated\/} if there is a finite subset of $M$ that generates $M$, {\it i.e.}, that is contained in no proper submonoid of $M$. 

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\noindent {\bf Definition.} Given monoids $M_\alpha$, $\alpha\in A$, the {\it product\/} $\prod M_\alpha$ the set of functions $x:A\to \cup M_\alpha; \alpha\mapsto x_\alpha$, with $x_\alpha\in M_\alpha$, where we define $x\ast y$ by $(x\ast y)_\alpha = x_\alpha \ast  y_\alpha$.  (The product of a pair of monoids, $M$ and $N$ is denoted $M\times N$.)

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\noindent {\bf Definition.} A {\it congruence\/} on a monoid $M$ is a submonoid of $M\times M$ that is also an equivalence relation on $M$.  If $C$ is a congruence on $M$, then the set of equivalence classes is denoted $M/C$.

\bigskip
\noindent {\bf 1.1.~Fact.}  Given any subset $\Gamma\subseteq M\times M$,
there is a smallest congruence containing $\Gamma$, namely
$$\langle\langle\Gamma\rangle\rangle:=\bigcap\{\,C\mid \hbox{$C\supseteq\Gamma$
and $C$ is a congruence}\,\}.$$
 

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\noindent {\bf 1.2.~Exercise.}  Let $C$ be a congruence on $M$.  Let $[x]$ denote the equivalence class of $x\in M$.  \item{1)} If $(x,x')\in C$ and $(y,y')\in C$, then by assumption, $(x\ast y, x' \ast  y')\in C$, hence $[x\ast y] = [x' \ast  y']$.  
\item{2)} By 1), $[x]\ast [y]:= [x\ast y]$ determines a well-defined operation on $M/C$.  
\item{3)} Moreover, $M/C$ is a monoid with respect to this operation. 

\bigskip
\noindent {\bf Definition.} Let $L$ and $M$ be monoids.  A function $\phi:L\to M$ is called a {\it (monoid) morphism\/} if $i$) $\phi(e_L) = e_M$, and $ii$) $\phi(x\ast _L y) = \phi(x)\ast _M\phi(y)$ for all $x,y\in L$.

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\noindent {\bf 1.3.~Exercise.}  Let $\phi:L\to M$ be a morphism.  Let $C:= \{\,(x,y)\in L\times L\mid \phi(x)=\phi(y)\,\}$.  Then $C$ is a congruence on $L$, and $\phi$ can be written as the composition $\phi=\iota\circ \pi$, where $\pi:L\to L/C;x\mapsto[x]$ and $\iota: L/C \to M; [x]\mapsto \phi(x)$.   Moreover, $\iota$ is an injective morphism and $\pi$ is a surjective morphism.

\bigskip
\noindent  {\bf 1.4.~Fact.}  Any group is a monoid, any group homomorphism is a monoid morphism, and any monoid morphism between groups is actually a group homomorphism.  More formally, the category of groups is a full subcategory of the category of monoids. 

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\noindent  {\bf 1.5.~Fact.}  With any monoid $M$ we can associate a group $G(M)$ and a monoid morphism $\gamma_M:M\to G(M)$ with the following universal mapping property: given any monoid morphism $\phi:M\to H$ with $H$ a group, there is a unique group homomorphism $\overline\phi:G(M)\to H$ such that $\overline\phi\circ\gamma_M = \phi$.

\bigskip
\noindent  {\it Sketch of proof.}  Let $F$ be the free group on $|M|$.  Any finite sequence $u = (u_1, \ldots, u_n)$ of elements of $|M|$ determines an element $u_F:=u_1\cdot u_2\cdots u_n \in F$ and an element $u_M:=u_1\ast_M u_2\cdots \ast_M u_n \in M$.  Let $N$ be the ideal of $F$ generated by the set $$\{\,u_F\cdot{v_F}^{-1}\mid u_M = v_M \,\},$$
where $u$ and $v$ range over all finite sequences of elements of $|M|$.  One can show that the composition $M\to F\to F/N$ is a monoid morphism.  Moreover, one can also show that this morphism has the universal mapping property specified for $\gamma_M$.  Indeed, $\phi:M\to H$ considered merely as a map fo sets induces a group homomorphism $\hat\phi:F\to H$, and one can show that $N\subseteq \ker \hat\phi$.  Hence, $\hat\phi$ induces a group homomorphism $\overline\phi:F/N\to H$.

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\noindent  {\bf 1.6. Exercise.}  Provide the details in the sketch.
\bigskip
\noindent  {\bf 1.7. Exercise.}  Show that $G(M)$ is commutative if $M$ is.  

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\noindent  {\bf Problem.}  Find (interesting) necessary and/or sufficient conditions for $\gamma_M$ to be injective.   {\it Comment.} This problem  is discussed at length in Chapter 12 (Volume II) of [CP].

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\noindent{\bf References}

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 \itemitem{[CP]} A.~H.~Clifford \& G.~B.~Preston, {\it The Algebraic Theory of Semigroups}, AMS 1967.



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