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A semigroup with a noncommutative product in which no product can ever be expressed more simply in terms of other elements.

A chain complex is a sequence of maps ...-->^(partial_(i+1))C_i-->^(partial_i)C_(i-1)-->^(partial_(i-1))..., (1) where the spaces C_i may be Abelian groups or modules. The ...

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups. The matrix direct sum is defined by ...

A short exact sequence of groups 0-->A-->B-->C-->0 (1) is called split if it essentially presents B as the direct sum of the groups A and C. More precisely, one can construct ...

Abel's integral is the definite integral I = int_0^infty(tdt)/((e^(pit)-e^(-pit))(t^2+1)) (1) = 1/2int_(-infty)^infty(tdt)/((e^(pit)-e^(-pit))(t^2+1)) (2) = ...

A square matrix A is antihermitian if it satisfies A^(H)=-A, (1) where A^(H) is the adjoint. For example, the matrix [i 1+i 2i; -1+i 5i 3; 2i -3 0] (2) is an antihermitian ...

An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity A=-A^(T) (1) where A^(T) is the matrix transpose. ...

Let A_1, A_2, A_3, and A_4 be four points on a circle, and H_1, H_2, H_3, H_4 the orthocenters of triangles DeltaA_2A_3A_4, etc. If, from the eight points, four with ...

The extangents circle is the circumcircle of the extangents triangle. Its center function is a complicated 9th-order polynomial and its circle function is a complicated ...

The reflection circle, a term coined here for the first time, is the circumcircle of the reflection triangle. It has center at Kimberling center X_(195), which is the ...