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If an algebraic structure is a set of operations on a set of elements, what is a non-algebraic structure?

https://en.wikipedia.org/wiki/Outline_of_algebraic_structures#Algebraic_structures_with_additional_non-algebraic_structure

Lance
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  • It's referring to structure that doesn't come from operations on the underlying set. For example, an order doesn't come from the operations on the underlying elements (but it might be compatible with the operations). –  Mar 05 '18 at 22:01

1 Answers1

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As soon as we allow relations, we've left the realm of algebras (in the universal algebra sense). For example, a linear ordering is a set $X$ together with a binary relation $<$ such that [stuff]; a linear ordering is definitely a set with structure, but that structure isn't given by functions, so it's not an algebra.

There are also structures that mix relation and function symbols, such as ordered groups/rings/fields. And something like a topological group is even further from this picture, since the "additional structure" (= the topology) exists at a "higher level" than just functions/relations on the underlying set.

Noah Schweber
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  • What about relational algebra? http://infolab.stanford.edu/~ullman/fcdb/aut07/slides/ra.pdf – Lance Mar 05 '18 at 22:10
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    @LancePollard A relational algebra is a structure whose elements are relations on a different set $X$. But this is different from endowing the set $X$ itself with relations. The set of relations on a set $X$ together with composition (and possibly other operations) forms an algebra, but the set $X$ equipped with a bunch of relations doesn't. They're just two different kinds of thing. – Noah Schweber Mar 05 '18 at 22:12
  • @NoahSchweber My understanding is that an equation may introduce order on the naturals that contains an addition operation which may be defined for that system with a successor function. Without that function, the equation introducing order on the set would contain an operation that lacks definition, and so it couldn't introduce the order. With this in mind, how might I clarify my understanding of what you meant by "that structure isn't given by functions"? – bblohowiak Feb 24 '19 at 15:00