homology

noun homologies

1. countable, uncountable specifically, such relationship in the context of the geometry of perspective. 1863, George Salmon, A Treatise on Conic Sections, Longman, Brown, Green, Longman, and Roberts, 4th Edition, page 61, Two triangles are said to be homologous, when the intersections of the corresponding sides lie on the same right line called the axis of homology: prove that the lines joining the co
2. countable, uncountable 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.
3. countable, uncountable 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. One encounters a similar situation in homology theory. Beside singular homology, which is a homotopy invariant, and Čech homology, which is a shape invariant, there exists strong homology, which is a strong shape invariant. In the special case of metric compacta, this homology was introduced by N.E.
4. countable, uncountable 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.
5. countable, uncountable The relationship, between elements, of being in the same group of the periodic table.
6. countable, uncountable The relationship, between organic compounds, of being in the same homologous series.
7. countable, uncountable The relationship, between characteristics or behaviours, of having a shared evolutionary or developmental origin; 2000, Julie A. Hawkins, Chapter 2: A survey of primary homology assessment, Robert Scotland, R. Toby Pennington (editors), Homology and Systematics, Taylor & Francis, The Systematics Association, page 22, The objective of this study is to classify approaches to primary homology assessment, and to qu

   Opposite: heterology

8. countable, uncountable specifically, a correspondence between structures in separate life forms having a common evolutionary origin, such as that between mammalian flippers and hands.

   Opposite: heterology

9. countable, uncountable The presence of the same series of bases in different but related genes.
10. countable, uncountable The relationship, between temporally separated human beliefs, practices or artefacts, of possessing shared characteristics attributed to genetic or historical links to a common ancestor.