Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Simple Search

Query:
Enter a query and click »Search«...
Format:
Display: entries per page entries
Zbl 0713.20033
Kolibiar, Milan
Median groups.
(English)
[J] Arch. Math., Brno 25, No.1-2, 73-82 (1989). ISSN 0044-8753; ISSN 1212-5059/e

By median algebra is meant an algebra with one ternary operation (,,) satisfying the identities $(a,a,b)=a$, $((a,d,c),b,c)=((b,c,d),a,c)$. By a median group (m-group) there is meant an algebra $(G;+,-,0,(,,))$ where $(G;+,-,0)$ is a group, (G;(,,)) is a median algebra and the identity $u+(a,b,c)+v=(u+a+v,u+b+v,u+c+v)$ holds. If ${\cal G}$ is an l-group then the m-group M(${\cal G}):=(G;+,-,0,(,,))$, where the ternary operation (,,) is given by $(a,b,c)=(a\wedge b)\vee (b\wedge c)\vee (c\wedge a)$ is said to be associated with ${\cal G}$. Such an m-group satisfies (*) $(x,0,-x)=0$. A subset L of a median algebra with card $L\ne 4$ is a line iff for any a,b,c$\in L$ one of the relations abc, bca or cab holds. (If $(a,b,c)=b$, we say that b is between a and c and denote it by abc.) The author deduces a number of results some of which are mentioned here. Theorem. Let an m-group ${\cal G}$ satisfy the identity (*) and let A be a line in M(${\cal G})$ such that $0\in A$. Then the following are equivalent (a) A forms a subgroup of $(G;+,-,0)$ and a direct factor of ${\cal G}$. (b) A is a convex maximal line of ${\cal G}$. (A is said to be convex if a,b$\in A$, $u\in G$, aub imply $u\in A.)$ Theorem. Let ${\cal G}$ be an m-group satisfying (*) and A be a convex maximal line in ${\cal G}$. If $a\in G$ then $-a+A$ is a direct factor in G.
[F.Šik]
MSC 2000:
*20F60 Ordered groups (group aspects)
20N10 Ternary systems (group theory)
08A02 General relational systems
06F15 Ordered groups
20F05 Presentations of groups
20E07 Subgroup theorems (group theory)

Keywords: median algebra; ternary operation; identities; median group; l-group; m- group; subgroup; direct factor; convex maximal line

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences