In his message of Mon, 04 Oct 2010 08:16:25 AM EDT, Vaughan Pratt <pratt@cs.stanford.edu> quibbled with what on 10/2/2010 3:03 PM, Michael Shulman had written:
I personally prefer to say that "unique choice structure" is something "in between" property and structure. Kelly and Lack dubbed it "Property-like structure" in their paper with that title. The difference is exactly as you say: property-like structure is unique (up to unique isomorphism) when it exists, but is not necessarily "preserved" by all morphisms.
How should this terminology be applied when the property-like structure is necessarily preserved by all morphisms?
A group can be defined as a monoid with the property that all of its elements have inverses. The inverse is preserved by all morphisms.
A group can also be defined as a *semigroup* with that property. "The inverse" need no longer be "preserved by all morphisms."
A Boolean algebra can be defined as a bounded distributive lattice with the property that all of its elements have complements. The complement is preserved by all morphisms.
Depends what you take to be a bounded lattice. Do you specify *finitary* meets and joins, including the explicit empty ones that produce the bounds? Or just *binary* ones, with the bounds *required* but not *specified*? In the former situation, yes, "the complement is preserved by all morphisms." In the latter situation, alas, no.
Are these merely "property-like structures," or are they actual structures, despite being defined merely as properties?
When such a "property-like structure" *is* preserved, it is perhaps implicitly trying to behave like an "actual" structure, and could certainly be harmlessly added to the actual structural specifications, but, with so much riding on the *context* in which one is asking about that property-like structure, I'm not yet ready just to declare them, willy-nilly, to be "actual structures". Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]