Categories of representations of a class of commutative cancellative semigroups (Q2773007)

Let \(S\) be a commutative semigroup. \(S\) is called subarchimedean if there exists \(z\in S\) such that for any \(a\in S\) there are \(n>0\) and \(x\in S\) such that \(z^n=ax\). A commutative cancellative idempotent free subarchimedean semigroup is called a \(\mathcal P\)-semigroup. (The reviewer and his students called these semigroups \(\overline{\mathcal N}\)-semigroups and studied them by the usual treatment [\textit{H. B. Hamilton, T. E. Nordahl, T. Tamura}, Pac. J. Math. 61, 441-456 (1975; Zbl 0358.20073)].) In this paper the authors perform a categorical treatment which nicely uniformly reorganizes and extends widely the results of the reviewer and his students.NEWLINENEWLINENEWLINEFirst, notations, \(P\) the set of positive integers, \(N\) nonnegative integers, \(Z\) integers, \(Q\) rational numbers. Let \(X\) be a set, \(\iota_X\) the identity map on \(X\), \(0_X\colon X\to 0\), \(1_X\colon X\to 1\). For \(\varphi\colon X\to Q\), define \(\varphi\pm 1\) by \(a(\varphi\pm 1)=a\varphi\pm 1\). Let \(A\), \(B\) be Abelian groups, if \(\theta\colon A\to B\), \((a,b)\theta=(a\theta)(b\theta)((ab)\theta)^{-1}\) for \(a,b\in A\). For \(I\colon G\times G\to Z\) and \(a\in G\), let \(aI_{-1}=aI_0=0\) and \(aI_n=\sum^n_{i=1}(a,a^i)I\) for \(n\in P\). In the construction below the following are assumed. NEWLINE\[NEWLINE\begin{matrix}{\l}\quad &{\l}\\ (R_N)\quad(a,b)\varphi\in N,&(R_Z)\quad(a,b)\varphi\in Z,\\ (N_1)\quad e\varphi=(e,e)I =1,&(N_0)\quad e\varphi=(e,e)I=0,\\ (C)\quad(a,b)I=(b,a)I,&(A)\quad(a,b)I+(ab,c)I=(a,bc)I+(b,c)I,\\ (M_\kappa)\quad(s,a)(t,b)=(s+t,ab),&(M_y)\quad(m,a)(n,b)=(m+n+(a,b)I,ab).\end{matrix}NEWLINE\]NEWLINE Let \(G\) be a group, \(I\colon G\times G\to N\) satisfying \((A)\), \((N_1)\); define a groupoid \(N(G,I)\) on the set \(N\times G\) by \((N_y)\). Let \(G\) be a group, \(I\colon G\times G\to Z\) with \((A)\) and \((N_0)\), define a groupoid \(Z(G,I)\) on the set \(Z\times G\) by \((M_y)\). Let \(G\) be an Abelian group, \(\varphi\colon G\to Q\) with \((R_N)\) and \((N_1)\), define \(N(G,\varphi)\) on \(\{(s,a)\in Q\times G\mid s-a\varphi\in N\}\) by \((M_\kappa)\). Let \(G\) be an Abelian group, \(\varphi\colon G\to Q\) with \((R_Z)\), \((N_0)\), and define \(Z(G,\varphi)\) on \(\{(s,a)\in Q\times G\mid s-a\varphi\in Z\}\) by \((M_y)\).NEWLINENEWLINENEWLINEThen the following is the theorem (1): Let \(G\) be a group, and \(I\colon G\times G\to N\) satisfying \((A)\) and \((N_1)\). Then \(N(G,I)\) is a cancellative idempotent-free subarchimedean semigroup having a central \(p\)-element. Conversely, every semigroup having these properties is isomorphic to some \(N(G,I)\). An element \(z\) of \(S\) is central if it commutes with all elements of \(S\), it is a \(p\)-element (pivot element) if for any \(a\in S\), there exist a positive integer \(n\) and \(x,y\in S\) such that \(z^n=ax=ya\). To define categories, we assume the following: NEWLINE\[NEWLINE\begin{matrix}{\l}\quad &{\l}\\ (C_N)\quad a\sigma-a\tau\psi\in N, &(C_Z)\quad a\sigma-a\tau\psi\in Z,\\ (\Sigma_N)\quad (a,b)\sigma=(e\sigma)(a,b)\varphi, &(\Sigma_Z)\quad(ab)\sigma=(e\sigma)[(a,b)\varphi+1],\\ (T_N)\quad(a,b)\tau=(e\tau)^{(a,b)\varphi}, &(T_Z)\quad(a,b)\tau=(e\tau)^{(a,b)\varphi+1},\\ (B_N)\quad(a,b)\beta=(e\beta)^{(a,b)I}, &(B_Z)\quad(a,b)\beta=(e\beta)^{(a,b)I+1}.\end{matrix}NEWLINE\]NEWLINE NEWLINE\[NEWLINE(A_N)\quad(a,b)\alpha=\begin{cases}(e\alpha)(a,b)I-(a\beta,b\beta)J+e\beta J_{(a,b)I-1}+((e\beta)^{(a,b)I},(ab)\beta)J &\text{if }(a,b)I>0\\ -(a\beta,b\beta)J &\text{if }(a,b)I=0\end{cases}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{multlined}(A_Z)\quad (a,b)\alpha=(e\alpha)[(a,b)I+1]-(a\beta,b\beta)J+((e\beta)^{(a,b)I+1},(ab)\beta)J+\\ +\begin{cases} e\beta J_{(a,b)I} &\text{if }(a,b)I\geq-1,\\ -e\beta J_{-(a,b)I-2}-((e\beta)^{(a,b)I+1},(e\beta)^{-(a,b)I-1})J &\text{if }(a,b)I<-1.\end{cases}\end{multlined}NEWLINE\]NEWLINE We define categories. Category \(\mathcal S\): \(\mathcal P\)-semigroups and their homomorphisms. Category \(\mathcal G\): Non-periodic Abelian groups and their homomorphisms. Category \(\mathcal X\): \(\text{Ob }{\mathcal X}\) is the set of \((G,\varphi;N)\), \(G\) Abelian group, \(\varphi\colon G\to Q\) with \((R_N)\) and \((N)\), and \(\Hom_{\mathcal X}((G,\varphi;N),(H,\psi;N))\) is the set of all pairs of functions \((\sigma,\tau)\) where \(Q\overset\sigma\leftarrow G\overset\tau\rightarrow N\) with \((C_N)\), \((\Sigma_N)\), \((T_N)\) and \(\text{id}_{(G,\varphi;N)}=(G,i_G)\). For \(\mathcal X\)-morphisms NEWLINE\[NEWLINE(G,\varphi;N)\overset{(\sigma,\tau)}\longrightarrow(G',\varphi';N)\overset{(\sigma',\tau')}\longrightarrow(G'',\varphi'';N).NEWLINE\]NEWLINE Let \((\sigma,\tau)(\sigma',\tau')=(\sigma'',\tau'')\) where \(a\sigma''=(e'\sigma')(a\sigma-a\tau\varphi')+a\tau\sigma'\), \(a\tau''=(e'\tau')^{a\sigma-a\tau\varphi'}(a\tau\tau')\).NEWLINENEWLINENEWLINECategory \({\mathcal X}'\) is defined from the description of the definition of \(\mathcal X\) by replacing \(N\) by \(Z\), and \(-a\tau\varphi'\) by \(-a\tau\varphi'-1\). Category \(\mathcal Y\): \(\text{Ob }{\mathcal Y}\) is the set of \((G,I,N)\), \(G\) Abelian group, \(I\colon G\times G\to N\) with \((A)\), \((C)\) and \((N_1)\); \(\Hom((G,I;N),(H,J;N))\) consists of all pairs \((\alpha,\beta)\) where \(N\overset\alpha\leftarrow G\overset\beta\rightarrow H\) with \((A_N)\) and \((B_N)\) and \(\text{id}_{(G,I;N)}=(0_G,\iota_G)\). For \(\mathcal Y\)-morphism \((G,I,N)\overset{(\sigma,\tau)}\longrightarrow(G',I';N)\overset{(\sigma',\tau')}\longrightarrow (G'',I'';N)\), let \((\alpha,\beta)(\alpha',\beta')=(\alpha'',\beta'')\) where NEWLINE\[NEWLINEa\alpha''=\begin{cases} a\beta\alpha'+(a\alpha)(e'\alpha')+e'\beta'I_{a\alpha-1}''+((e'\beta')^{a\alpha},a\beta\beta')I'' &\text{if }a\alpha>0,\\ a\beta\alpha' &\text{if }a\alpha=0.\end{cases}NEWLINE\]NEWLINE NEWLINE\[NEWLINEa\beta''=(e'\beta)^{a\alpha}(a\beta\beta').NEWLINE\]NEWLINE Functor \(Q\): For \(S\), let \(SQ=(S\times S)/\sim\) where \((a,b)\sim(c,d)\) iff \(ad=bc\), \([a,b][c,d]=[ac,bd]\). \(SQ\) is the group of quotients of \(S\). For \(\psi\colon S\to T\) let \(\psi Q\colon[a,b]\mapsto[a\psi,b\psi]\) for \([a,b]\in SQ\). The mapping \(\delta\colon s\to[s^2,s]\).NEWLINENEWLINENEWLINEFunctor \(X\): For \((G,\psi;N)\in\text{Ob }{\mathcal X}\), let \((G,\varphi;N)X=(G,\varphi-1;Z)\) and let \(X\) be the identity map on \(\Hom{\mathcal X}\).NEWLINENEWLINENEWLINEFunctor \(\Gamma\): For \((G,\varphi;N)\in\text{Ob }{\mathcal X}\), let \((G,\varphi;N)\Gamma=(G,I;N)\) where \(I\colon(a,b)\mapsto(a,b)\varphi\) for \(a,b\in G\). For \(\mathcal X\)-morphism \((\sigma,\tau)\colon(G,\varphi;N)\to(G,\psi;N)\), let \((\sigma,\tau)\Gamma=(\alpha,\tau)\) where \(\alpha\colon a\mapsto a\sigma-a\tau\psi\) for \(a\in G\).NEWLINENEWLINENEWLINEFunctor \(\Phi\): For \((G,\varphi;N)\in\text{Ob }{\mathcal X}\), let \((G,\varphi;N)\Phi=N(G,\varphi)\). For an \(\mathcal X\)-morphismNEWLINENEWLINENEWLINE\((\sigma,\tau)\colon(G;\varphi;N)\to(H,\psi;N)\), let NEWLINE\[NEWLINE(\sigma,\tau)\Phi\colon(s,a)\mapsto((ea)(s-a\varphi)+a\sigma,(e,\tau)^{s-a\varphi}(a\tau))\text{ for }(s,a)\in N(G,\varphi).NEWLINE\]NEWLINE Some of the main theorems of this paper other than (1) are as follows: (2) The groupoid \(N(G,\varphi)\) is a \(\mathcal P\)-semigroup generated by the set \(\{(a\varphi,a)\mid a\in G\}\) where \((1,e)\) is a pivot element. Conversely every \(\mathcal P\)-semigroup is isomorphic to some \(N(G,\varphi)\).NEWLINENEWLINENEWLINE(3) The functor \(Q\colon{\mathcal S}\to{\mathcal G}\) is faithful and representative.NEWLINENEWLINENEWLINE(4) The functor \(\Phi\colon{\mathcal X}\to{\mathcal S}\) is an equivalence of categories and is injective on objects.NEWLINENEWLINENEWLINE(5) The functor \(X\colon{\mathcal X}\to{\mathcal X}'\) is a representative embedding but is not full.NEWLINENEWLINENEWLINE(6) The functor \(\Gamma\colon{\mathcal X}\to{\mathcal Y}\) is an equivalence of categories and is surjective on objects.NEWLINENEWLINENEWLINEThe reviewer respects the authors since the results of the reviewer and his students of 1975 are again in the limelight by means of modern ideas.