# Why are structures with no relations called algebras?

"If [a given structure] A has no relations it is termed an algebraic structure, or simply an algebra" -

Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, page 42.

I didn't understand Rautenberg's definition. Elementary algebra has at least one relation: the equality (or identity) relation, signalized by the symbol "="

The equality relation is quintessential to linear algebra and algebraic equations, such as "x +5 = y"

As linear algebra and algebraic equations are written in a language whose meaning is given by a structure and an interpretation function, why are structures with no relations called algebras?

## 1 Answer

Equality is a logical symbol not a relation of the language.

The difference is the following. A structure is free to interprets relation symbols in an arbitrary way (just the arity counts). On the other hand the definition of structure requires that equality is interpreted as the the "real" equality (not just an equivalence/congruence relation).

• Treating equality as a logical symbol is indeed by far the most common convention, but it is not universally adopted - e.g., it is not unusual to present equality as a defined notion in axiomatisations of set theory. However, if someone writes that a structure has no relations, then they must be classing equality as a logical symbol, because otherwise the first-order language of the structure would be empty - there would be no atomic predicates. – Rob Arthan Apr 23 '15 at 19:39