Is there a definition of algebraic structure? Wikipedia says:
a set (called carrier set or underlying set) with one or more finitary operations defined on it.
In particular, what is the motivation to say that a lattice is an algebraic structure, but not saying that a poset is an algebraic structure? Are they too boring? Referring to wikipedia, the term "one or more finitary operations" seems to hold for almost everything I know.
On
I know that in literature is far easier to find definitions of magma, groupoid, binary operation, etc., rather than a definition of algebraic structure.
One that may fit is the following.
An algebraic structure is a totally ordered set, whose elements are
- sets $S_i$,
- (finitary) operations $O_j$ over these sets,
- relations $R_k$ between these sets.
I wish I'll add a reference soon (sorry for only translating this page back from my native language, italian).
Related, this definition includes relations, as the above. And the bibliography at the bottom of this latter may give you a further hand.
An example of structure with more than one set is a vector space $\mathrm{W}$ over a field $\Bbb K$: $(\mathbb{K},\mathrm{W},+,\times)$.
An example of structure with also a relation is an ordered field: $(\mathbb{K},+,\cdot,\le)$.
A poset has a structure, but not an algebraic structure. A poset is equipped with a partial order relation, this is not a finitary operation. For a set $S$ an $n$-tary operation is a function $F:S^n\to S$. An order relation is not a function.