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{\hfill{Fall 2021} \smallskip %
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{\LARGE Introduction to Algebraic Structures\\} 

{\LARGE Problem Set 6\\}
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{\LARGE Acts over monoids.}
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\noindent 1. Let $B\subset A$ be two sets and consider the monoid $(\mathcal{P}(B),\cap)$. Are the maps $\cap,\cup:\mathcal{P}(A)\times \mathcal{P}(B)\to\mathcal{P}(A)$,  $(X,Y)\mapsto X\cap Y$ and $(X,Y)\mapsto X\cup Y$, $X\subset A, Y\subset B$ monoid actions of $\mathcal{P}(B)$ on $\mathcal{P}(A)$?

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\noindent 2. Give an example of a \emph{noncommutative} monoid $S$ and an \emph{infinite} set $A$ with left and right $S$-actions so that the two actions are \\\indent a) as similar as possible; b) as dissimilar as possible.

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\noindent 3. Let $T:=\mathcal{T}_{\{1,2,3\}}$ be the full transformation monoid of the set $\{1,2,3\}$, $f,g\in T$, $f(1)=f(2)=3$, $f(3)=1$; $g(1)=2$, $g(2)=g(3)=1$. Find the \mbox{smallest} subact of $_TT$ that contains both $f$ and $g$. Composition is from right to left. 

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\noindent 4. Find all endomorphisms of the \emph{set} $\Z$ considered as an act over the \emph{monoid} $(\Z,+)$.

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\noindent 5. Consider $\N$ as a right act over $(\N,\max)$ with the action $$n\cdot m = \min\{p\in\P\mid p\geq n\}.$$ Write $\N_\N$ as a disjoint union of indecomposable subacts. 

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\noindent 6. Prove that an $S$-act $F_S$ is free with basis $X$ if and only if for any $S$-act $A_S$ and every map $f:X\to A$ there is a \emph{unique} homomorphism \mbox{$g:F_S\to A_S$} with the property $f=g\iota$, where $\iota: X\to F_S$ is the canonical insertion. 

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\noindent 7. Consider $S=[0,1]\times[1,\infty)$ as a monoid with multiplication defined by $(a,b)(c,d)=(\min(a,c),\max(b,d))$. Find all projective acts over $S$. 

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\noindent 8. Consider the monoid $N=(\N\cup\{0\},+)$ and $\Z$ as an $N$-act with the action $(a,n)\mapsto a+n$. Find all subacts and all quotient acts of $\Z_N$. 

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\noindent 9${^*}$. Find all subacts and all quotient acts of the monoid $(\N,\cdot)$ considered as an act over itself.

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