homology

noun11

1. •
   The relationship of being homologous; a homologous relationship.
2. •
   specifically, such relationship in the context of the geometry of perspective.

   • Two corresponding straight lines therefore always intersect on a fixed straight line, which we may call s; thus the given figures are in homology, O being the centre, and s the axis, of homology. — 1885
3. •
   The relationship, between elements, of being in the same group of the periodic table.
4. •
   The relationship, between organic compounds, of being in the same homologous series.
5. •
   The relationship, between characteristics or behaviours, of having a shared evolutionary or developmental origin;
6. •
   specifically, a correspondence between structures in separate life forms having a common evolutionary origin, such as that between mammalian flippers and hands.
7. •
   The presence of the same series of bases in different but related genes.
8. •
   The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.
9. •
   A general way of associating a sequence of algebraic objects, such as abelian groups or modules, to a sequence of topological spaces; also used attributively: see Usage notes below.

   • Brieskorn homology spheres are a special case of Siefert fibered homology spheres. — 2002
10. •
    Given a chain complex {Gₙ} and its associated set of homomorphisms {Hₙ}, the rule which explains how each Hₙ maps Gₙ into the kernel of Gₙ₊₁.

    • Because of their connection with both homology and cohomology, chain complexes are an important topic of study in homological algebra.
11. •
    An automorphism of the projective plane (representing a perspective projection) that leaves all the points of some straight line (the homology axis) fixed and maps all the lines through some single point (the homology centre) onto themselves.

    • If the homology centre lies on the homology axis, the homology is said to be singular or parabolic; otherwise, it is called non-singular or hyperbolic.