An abelian group is a structure G = (G, + ,−,0), where + is an infix binary operation, called the
group addition, − is a prefix unary operation, called the
group negative and 0 is a constant (nullary operation), called the additive identity element, such that
+ is commutative: x + y = y + x,
+ is associative: (x + y) + z = x + (y + z),
0 is an additive identity for + : 0 + x = x, and
− gives an additive inverse for + : −x + x = 0.
Let G and H be abelian groups. A morphism from G to H is a function h : G→H that is a homomorphism: h(x + y) = h(x) + h(y).
Remark: It follows that h(−x) = −h(x) and h(0) = 0.
(Z, + , −, 0), the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra.
|Equational theory||decidable in polynomial time|
|First-order theory||decidable [W. Szmielew, Decision problem in group theory, Library of the Tenth International Congress of Philosophy, Amsterdam, August 11--18, 1948, Vol.1, Proceedings of the Congress (1949) 763--766 MRreview]|
|Congruence distributive||no (Z2×Z2)|
|Congruence n-permutable||yes, n = 2, p(x,y,z) = x−y + z|
|Congruence regular||yes, congruences are determined by subalgebras|
|Congruence extension property||yes, if K ≤ H ≤ G then K ≤ G|
|Definable principal congruences||no|
|Equationally definable principal congruences||no|
|Strong amalgamation property||yes|
|Epimorphisms are surjective||yes|