On monoids in the category of sets and relations. (English) Zbl 1387.81022
Summary: The category \(\mathbf{Rel}\) is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, \(\mathbf{Rel}\) is a monoidal category. Moreover, \(\mathbf{Rel}\) is a locally posetal 2-category, since every homset \(\mathbf{Rel}(A,B)\) is a poset with respect to inclusion. We examine the 2-category of monoids \(\mathbf{RelMon}\) in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.
MSC:
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
| 06C15 | Complemented lattices, orthocomplemented lattices and posets |
| 06F05 | Ordered semigroups and monoids |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
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