If X is a groupoid, then for any \(x, y\in X\) we define \(\displaystyle \varPhi \,(x, y)=\big \{u\in X: \ \ u\,x=y\,\big \} \ \qquad \text{and}\qquad \ \varPsi \,(x, y)=\big \{v\in X: \ \ x\,v=y\,\big \}, \) and moreover \(\varphi \,(x)=\varPhi \,(x, x)\) and \(\psi \,(x)=\varPsi \,(x, x)\) . In the sequel, \(\varphi , \psi \) and \(\varPhi , \varPsi \) will be considered as relations on X and \(X^{2}\) to X, respectively. And, they will be called the main solvability relations in X. To feel the importance of these relations, note that if \(u\in \varphi \,(x)\) and \(v\in \varPsi \,(x, u)\) , then \(u\,x=x\) and \(x\,v=u\) . Therefore, u is a left unit for x and v is a right inverse of x relative to u. Thus, the above solvability relations can be used to classify and investigate groupoids. For instance, the groupoid X may be called prefunctional if the restrictions of the relations \(\varphi \) and \(\psi \) to the set \(\displaystyle X_{0}= \left \{ \begin {array}{ll} X\setminus \{0\} \ \ &\text{if }\ \ X\ \text{has a zero 0}; \\ \ \quad X &\text{if }\ \ X\ \text{does not have a zero} \end {array} \right . \) are functions of \(X_{0}\) to X. That is, \(\varphi \,(x)\) and \(\psi \,(x)\) are singletons for all \(x\in X_{0}\) . If X is a prefunctional groupoid, then by identifying singletons with their elements we may also define \(\displaystyle {\sigma \,(x)=\varPhi \,\big (x, \psi \,(x)\big ) \qquad \qquad \text{and}\qquad \qquad \rho \,(x)=\varPsi \,\big (x, \varphi \,(x)\big )} \) for all \(x\in X_{0}\) . Moreover, we may call the prefunctional groupoid X to be semifunctional if the relations \(\sigma \) and \(\rho \) are also functions of \(X_{0}\) to X. Surprisingly, if X is a prefunctional semigroup, then \(\sigma =\rho \) , and thus \(\rho \) is not needed. If X is a semifunctional semigroup with zero 0, then X may be called a Brand–Clifford semigroup and \(X_{0}\) may be called a Brand partial groupoid. Thus, the difficult definitions and properties of Brandt partial groupoids can be briefly expressed in terms of the solvability relations. The principal task is to determine the solvability relations in a given semigroup X. Unfortunately, this can be done only in some very particular cases. Of course, if X is a group, then the solvability relations in X can be easily computed, and are functions. In this respect it is also worth mentioning that a groupoid X may be called a quasi-group if it is functional in the sense that the relations \(\varPhi \) and \(\varPsi \) are functions of \(X^{2}\) to X. Moreover, the famous Green relations \(\mathscr {L}\) and \(\mathscr {R}\) can also be nicely defined in terms of the relations \(\varPhi \) and \(\varPsi \) considered in the groupoid obtained from X by adjoining a unit if necessary.

Solvability Relations in Groupoids

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2026年

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If X is a groupoid, then for any \(x, y\in X\) we define \(\displaystyle \varPhi \,(x, y)=\big \{u\in X: \ \ u\,x=y\,\big \} \ \qquad \text{and}\qquad \ \varPsi \,(x, y)=\big \{v\in X: \ \ x\,v=y\,\big \}, \) and moreover \(\varphi \,(x)=\varPhi \,(x, x)\) and \(\psi \,(x)=\varPsi \,(x, x)\) . In the sequel, \(\varphi , \psi \) and \(\varPhi , \varPsi \) will be considered as relations on X and \(X^{2}\) to X, respectively. And, they will be called the main solvability relations in X. To feel the importance of these relations, note that if \(u\in \varphi \,(x)\) and \(v\in \varPsi \,(x, u)\) , then \(u\,x=x\) and \(x\,v=u\) . Therefore, u is a left unit for x and v is a right inverse of x relative to u. Thus, the above solvability relations can be used to classify and investigate groupoids. For instance, the groupoid X may be called prefunctional if the restrictions of the relations \(\varphi \) and \(\psi \) to the set \(\displaystyle X_{0}= \left \{ \begin {array}{ll} X\setminus \{0\} \ \ &\text{if }\ \ X\ \text{has a zero 0}; \\ \ \quad X &\text{if }\ \ X\ \text{does not have a zero} \end {array} \right . \) are functions of \(X_{0}\) to X. That is, \(\varphi \,(x)\) and \(\psi \,(x)\) are singletons for all \(x\in X_{0}\) . If X is a prefunctional groupoid, then by identifying singletons with their elements we may also define \(\displaystyle {\sigma \,(x)=\varPhi \,\big (x, \psi \,(x)\big ) \qquad \qquad \text{and}\qquad \qquad \rho \,(x)=\varPsi \,\big (x, \varphi \,(x)\big )} \) for all \(x\in X_{0}\) . Moreover, we may call the prefunctional groupoid X to be semifunctional if the relations \(\sigma \) and \(\rho \) are also functions of \(X_{0}\) to X. Surprisingly, if X is a prefunctional semigroup, then \(\sigma =\rho \) , and thus \(\rho \) is not needed. If X is a semifunctional semigroup with zero 0, then X may be called a Brand–Clifford semigroup and \(X_{0}\) may be called a Brand partial groupoid. Thus, the difficult definitions and properties of Brandt partial groupoids can be briefly expressed in terms of the solvability relations. The principal task is to determine the solvability relations in a given semigroup X. Unfortunately, this can be done only in some very particular cases. Of course, if X is a group, then the solvability relations in X can be easily computed, and are functions. In this respect it is also worth mentioning that a groupoid X may be called a quasi-group if it is functional in the sense that the relations \(\varPhi \) and \(\varPsi \) are functions of \(X^{2}\) to X. Moreover, the famous Green relations \(\mathscr {L}\) and \(\mathscr {R}\) can also be nicely defined in terms of the relations \(\varPhi \) and \(\varPsi \) considered in the groupoid obtained from X by adjoining a unit if necessary.