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Two elements x and y of a set S are said to be commutative under a binary operation * if they satisfy x*y=y*x. (1) Real numbers are commutative under addition x+y=y+x (2) and ...

A compact surface is a surface which is also a compact set. A compact surface has a triangulation with a finite number of triangles. The sphere and torus are compact.

For {M_i}_(i in I) a family of R-modules indexed by a directed set I, let sigma_(ij):M_i->M_j i<=j be an R-module homomorphism. Call (M_i,sigma_(ij)) a direct system over I ...

A three-dimensional shadow of a four-dimensional Klein bottle.

The kernel of a group homomorphism f:G-->G^' is the set of all elements of G which are mapped to the identity element of G^'. The kernel is a normal subgroup of G, and always ...

A term used in category theory to mean a general morphism. The term derives from the Greek omicronmuomicron (omo) "alike" and muomicronrhophiomegasigmaiotasigma (morphosis), ...

The inversion of a horn torus. If the inversion center lies on the torus, then the horn cyclide degenerates to a parabolic horn cyclide.

A generalization of an ordinary two-dimensional surface embedded in three-dimensional space to an (n-1)-dimensional surface embedded in n-dimensional space. A hypersurface is ...

A three-dimensional shadow of a four-dimensional Klein bottle.

"The integers" is a common way of referring to the set of integers, commonly denoted Z.