Re: Semigroups with many objects

[Lutz Schroeder <lschrode@tzi.de>]

Dear all,

> I had also seen the word "precategory" but I cannot remember where. Beware of
> the fact that the word precategory is also used for categories *with
> identities* such that the composition law is partially defined : that is the
> fact that the codomain of F is equal to the domain of G is not sufficient for
> GoF to exist. Once again, I cannot remember where I read this word. The only
> thing I remember is that that was a computer-scientific work.

That would have been my paper with Paulo Mateus "Universal aspects of
probabilistic automata" in MSCS (and also "Monads on composition graphs"
in APCS). We do indeed use the word "precategory" for strucures with
identities, and with a partially defined composition law satisfying the
identity laws (strongly) and the associative law in the sense that
f(gh)=(fg)h holds strongly (or Kleene) provided that both gh and fg are
defined.

Moreover, as pointed out in a previous message, I have used the word
"semicategory" for similar structures, but with a stronger associative
law, requiring that f(gh)=(fg)h are both defined whenever fg and gh are
defined (or slight variations of this). Ehresmann used the term
"multiplicative graph" (and also sometimes "neocategory", I believe) for
structures satisfying the identity law, with no associativity imposed at
all.

-- Lutz


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Lutz Schroeder                  Phone +49-421-218-4683
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