# Isotopy (geometry)

In synthetic geometry, isotopy is a weakening of the isomorphism (of bodies and oblique bodies ) for ternary bodies (and more specific generalized bodies such as quasi-bodies and half-bodies ). The term isotopy takes into account the fact that for non-Desargue projective planes the algebraic structure of the coordinate range of the plane is generally not clearly defined by its geometric structure “except for isomorphism”. However, it could be shown that the coordinate ternary bodies of two projective planes, which are geometrically isomorphic, are always algebraically isotopic and that conversely projective planes, which can be coordinated by isotopic ternary bodies, are always geometrically isomorphic. Only with the projective relatives developed by Hans-Joachim Arnold from geometric relational algebra do the transition processes of algebraization and geometry return synonymously, i.e. H. except for isomorphism, in his projective classification theorem. In analogy to the corresponding terms derived from isomorphism , one speaks of isotopic generalized bodies if a triple of reversible mappings with certain structure-preserving properties exists between these solids, and then the mapping triple is called an isotopism .

→ Isotopisms of ternary bodies are a special case of isotopisms of quasi-groups . See also quasi-group # morphisms .

## definition

They are ternary bodies. A triple of 3 bijective mappings is called an isotopism from K to L , if ${\ displaystyle (K, T, 0_ {K}, 1_ {K}); (L, S, 0_ {L}, 1_ {L})}$${\ displaystyle (F, G, H)}$${\ displaystyle F, G, H \ colon K \ rightarrow L}$

1. ${\ displaystyle H (0_ {K}) = 0_ {L}}$ and
2. ${\ displaystyle H (T (a, b, c)) = S (F (a), G (b), H (c))}$ for all ${\ displaystyle a, b, c \ in K}$

applies. If there is an isotopism from K to L , the two ternary bodies are called isotopic to each other and they are called the same except for isotopy .

## properties

• Since the required mappings must be bijective, two isotopic ternary bodies are always equally powerful , if one of the isotopic ternary bodies is finite, so is the other, and their order (number of their elements) is the same.
• Two isomorphic ternary bodies are always isotopic: if there is an isomorphism, then there is an isotopism.${\ displaystyle H}$${\ displaystyle (H, H, H)}$
• For an isotopism always applies , whereby the “left” and the “right multiplication” are in the respective images of the one element.${\ displaystyle (F, G, H): K ^ {3} \ rightarrow L ^ {3}}$${\ displaystyle H = \ rho \ circ F = \ lambda \ circ G}$${\ displaystyle \ lambda: L \ rightarrow L; x \ mapsto S (F (1_ {K}), x, 0_ {L})}$${\ displaystyle \ rho: L \ rightarrow L; x \ mapsto S (x, G (1_ {K}), 0_ {L})}$${\ displaystyle L}$
• It is a planar ternary ring, . It should be left multiplication with and right multiplication with , . On is defined by a new ternary link . Then there is also a ternary body with the one element and is an isotopism.${\ displaystyle (K, T, 0,1)}$${\ displaystyle r, l \ in K \ setminus \ {0 \}}$${\ displaystyle \ lambda: K \ rightarrow K; x \ mapsto T (l, x, 0)}$${\ displaystyle l}$${\ displaystyle \ rho: K \ rightarrow K; x \ mapsto T (x, r, 0)}$${\ displaystyle r}$${\ displaystyle F_ {r}: = \ rho ^ {- 1}; G_ {l}: = \ lambda ^ {- 1}}$${\ displaystyle K}$${\ displaystyle S (a, b, c): = T (F_ {r} (a), G_ {l} (b), c)}$${\ displaystyle S}$${\ displaystyle (K, S, 0, \ epsilon)}$${\ displaystyle \ epsilon = T (l, r, 0)}$${\ displaystyle (F_ {r}, G_ {l}, \ mathrm {id} _ {K})}$
• If a ternary body, then every ternary body that is too isotopic is isomorphic to a ternary body that arises from one of the isotopisms so defined .${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle (F_ {r}, G_ {l}, \ mathrm {id} _ {K})}$
• For a finite ternary body of the order there are at most isotopic ternary bodies except for isomorphism .${\ displaystyle K}$${\ displaystyle n}$${\ displaystyle (n-1) ^ {2}}$
• The mentioned limit cannot be generally improved: In fact, there is a ternary body with 32 elements that is isotopic to it but has ternary bodies that are not isomorphic in pairs.${\ displaystyle 31 ^ {2}}$

### Isotopy is weaker than isomorphism

Two algebraic structures that are isomorphic to one another satisfy equally strong algebraic axioms. This generally no longer applies to ternary bodies if they are only isotopic to one another. If one introduces an addition and a multiplication , as described in the article Ternary Fields , with which the ternary connection in linear ternary fields can be represented as , then some ternary fields fulfill stronger axioms for generalized fields. The following applies: ${\ displaystyle T (a, b, c) = a \ cdot b + c}$${\ displaystyle (K, +, \ cdot, 0,1)}$

• A body or oblique body is isotopic to a ternary body if and only if it is isomorphic to it. In particular, the isotopic image of a body is again a body and that of an oblique body is an oblique body.
• If a half body is isotopic to a ternary body , then it is also a half body. For every projective plane that is not a Moufang plane , there are no isomorphic coordinate areas. Therefore, for every true half body there is an isotopic, but not isomorphic half body.${\ displaystyle L}$${\ displaystyle L}$
• If a quasi-body is isotopic to a ternary body , then in general there is no need to be a quasi-body. For every finite quasi-body that is not a half-body, there is an isotopic ternary body that is not a quasi-body.${\ displaystyle L}$${\ displaystyle L}$

## literature

5. This is also counted. The count is based on the possibilities of choosing the elements and from , Knuth (1963), Theorem 3.2.3${\ displaystyle K}$${\ displaystyle r}$${\ displaystyle l}$${\ displaystyle K \ setminus \ {0 \}}$