atoms below it). The quotient algebras Bi of B are defined inductively by B0=B, Bk+1 = Bk/I(Bk). The invariant m(B) is the smallest integer such that Bm+1 is trivial, or ∞ if no such integer exists. If m(B) is finite, the invariant n(B) is the number of atoms of Bm(B) if this number is finite, or ∞ if this number is infinite. The invariant l(B) is 0 if Bm(B) is atomic or if m(B) is ∞, and 1 otherwise. Then two Boolean algebras are elementarily equivalent if and only if their invariants l, m, and n are the same. In other words, the values of these invariants classify the possible completions of the theory of Boolean algebras. So the possible complete theories are: The trivial algebra (if this is allowed; sometimes 0≠1 is included as an axiom.) The theory with m = ∞ The theories with m a natural number, n a natural number or ∞, and l = 0 or 1 (with l = 0 if n = 0). Groups The signature of group theory has one constant 1 (the identity), one function of arity 1 (the inverse) whose value on t is denoted by t−1, and one function of arity 2, which is usually omitted from terms. For any integer n, tn is an abbreviation for the obvious term for the nth power of t. Groups are defined by the axiomsIdentity: ∀x 1x = x ∧ x1 = xInverse: ∀x x−1x = 1 ∧ xx−1 = 1Associativity: ∀x∀y∀z (xy)z = x(yz) Some properties of groups that can be defined in the first-order language of groups are: Abelian: ∀x ∀y xy = yx. Torsion free: ∀x x2 = 1→x = 1, ∀x x3 = 1 → x = 1, ∀x x4 = 1 → x = 1, ... Divisible: ∀x ∃y y2 = x, ∀x ∃y y3 = x, ∀x ∃y y4 = x, ... Infinite (as in identity theory) Exponent n (for any fixed positive integer n): ∀x xn = 1 Nilpotent of class n (for any fixed positive integer n) Solvable of class n (for any fixed positive integer n) The theory of abelian groups is decidable. The theory of infinite divisible torsion-free abelian groups is complete, as is the theory of infinite abelian groups of exponent p (for p prime). The theory of finite groups is the set of first-order statements in the language of groups