On intrinsic quasimetrics preserving maps on non-abelian partially ordered groups

Rok, strany: 2004, 135 - 140

In [JASEM, M.: Intrinsic metric preserving maps on partially ordered groups, Algebra Universalis 36 (1996), 135–140], it was proved that a stable surjective map $f$ from an abelian directed group $G₁$ onto a directed group $G₂$ is a homomorphism if it satisfies the following condition: \roster \item"(C)" If $| x - y | = | z - t |$, then $| f(x)-f(y) | = | f(z) - f(t) |$ for each $x, y, z, t \in G₁$. \endroster In this paper a stable map $f: G₁ \to G₂$ satisfying \thetag{C} is studied, where $G₁$ and $G₂$ are non-abelian directed groups. It is shown that a stable injective map $f:G₁\to G₂$ satisfying \thetag{C} is a homomorphism in the case that $G₁$ is a $2$@-isolated directed group and $G₂$ is a linearly ordered group. The question whether $f$ is a homomorphism also in the case of non-linearly ordered group $G₂$ remains open.

ISO 690:

Jasem, M. 2004. On intrinsic quasimetrics preserving maps on non-abelian partially ordered groups. In Mathematica Slovaca, vol. 54, no.3, pp. 135-140. 0139-9918.

APA:

Jasem, M. (2004). On intrinsic quasimetrics preserving maps on non-abelian partially ordered groups. Mathematica Slovaca, 54(3), 135-140. 0139-9918.