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The absolute difference of two numbers n_1 and n_2 is |n_1-n_2|, where the minus sign denotes subtraction and |x| denotes the absolute value.

A mathematical object S is said to be additively closed if a,b in S implies that a+b in S.

If a map f:G->G^' from a group G to a group G^' satisfies f(ab)=f(b)f(a) for all a,b in G, then f is said to be an antihomomorphism. Moreover, if G and G^' are isomorphic, ...

An operator * for which a*b=-b*a is said to be anticommutative.

For operators A^~ and B^~, the anticommutator is defined by {A^~,B^~}=A^~B^~+B^~A^~.

If a map f:G->G^' from a group G to a group G^' satisfies f(ab)=f(b)f(a) for all a,b in G, then f is said to be an antihomomorphism.

The antilaplacian of u with respect to x is a function whose Laplacian with respect to x equals u. The antilaplacian is never unique.

A quantity which changes sign when indices are reversed. For example, A_(ij)=a_i-a_j is antisymmetric since A_(ij)=-A_(ji).

A valuation for which |x|<=1 implies |1+x|<=C for the constant C=1 (independent of x). Such a valuation does not satisfy the strong triangle inequality |x+y|<=max(|x|,|y|).

A group in which any decreasing chain of distinct subgroups terminates after a finite number.