I am concerned with the distinction between models of first order logic and models of equational logic. The term model seems to be a near synonym for structure.
Joseph Goguen writes:
"Many logical systems have been shown to be institutions, including first order logic (with first order structures as models, denoted FOL), many sorted equational logic (with abstract algebras as models, denoted EQL)…"
I take the term abstract algebra to be the study of algebraic structures.
Definitions
Wikipedia says:
The structure consists of a domain of discourse D and an interpretation function I mapping non-logical symbols to predicates, functions, and constants.
Wikipedia says:
In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy
What are the distinctions between "first order structures" and "algebraic structures".
It seems to me that "algebraic structures" are restricted to equations in both the logic and the models. If we add equality to first order logic (FOLEQ) is an algebraic structure then just a particular type of FOLEQ structure?
Best Answer
A language (or vocabulary, or signature) $L$ is a set of constant, function, and relation symbols, with arities in $\mathbb{N}$ assigned to each function symbol and each relation symbol. (Note that it is common to treat constant symbols as function symbols of arity $0$.)
Given a language $L$, an $L$-structure is a set (sometimes required to be non-empty) equipped with interpretations of the constant, function, and relation symbols in $L$.
Let's say a language $L$ is an algebraic language if it has no relation symbols, and an algebraic structure is an $L$-structure for an algebraic language $L$.
The main thing I want to stress in this answer is that the definition of "structure" is separate from any considerations of logic. Note that I did not mention any axioms/identities/formulas in the above definitions.
Yes, an algebraic structure is a special kind of structure (one with no relation symbols in the language). We can consider algebraic structures and general $L$-structures in the context of any logic we want (equational logic, first-order logic with or without equality, infinitary logic, higher-order logic, etc.).
Also, as an aside: You write "If we add equality to first order logic...". The modern convention is that first-order logic includes equality by default. The version without equality is called "first-order logic without equality", but it's much less common. "Model theory" usually means model theory of first-order logic (with equality).
Usually, the study of arbitrary algebraic structures (especially equationally axiomatizable classes) is called "universal algebra" (or general algebra).
Both model theory and universal algebra are concerned with $L$-structures. It's just that in model theory we allow relation symbols in our languages, while universal algebra is (usually) concerned only with algebraic languages.
The real difference between model theory and universal algebra happens once we bring in the logic. Model theory concerns itself with first-order logic (classes of $L$-structures axiomatizable in first-order logic and first-order definability in those structures), while universal algebra concerns itself with equational logic (classes of $L$-structures for algebraic languages $L$ axiomatizable in equational logic and term operations in those structures).
If we have a theory $T$, a model of $T$ is a structure in which all the axioms of $T$ are true. People often use the word "model" loosely, referring to any structure as a "model". But it's more correct to only say "model" when there is a theory $T$ in the background.
Wikipedia is incorrect here. Identities or axioms are not part of the data of a structure (and equational theories, i.e. sets of identities, are not required to be finite). Again, the definition of "(algebraic) structure" is separate from any considerations of logic.