Monoids and Groups
نویسنده
چکیده
1 is unique (1 = 11 = 1, so any left identity equals any right identity). A morphism takes 1 to the identity of φX (since φ(x) = φ(1x) = φ(1)φ(x)), but φ(1) = 1 is an extra property that ought to be satisfied by morphisms; its kernel is the sub-algebra φ(1); zero object is { 1 }; not Cartesian-closed (the terminal and initial objects are the same, yet not all groups are isomorphic). A quasi-group is a set with an operation that satisfies cancelation: every equation ax = b, and xa = b, has a unique solution; a loop is a quasi-group with identity.
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