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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

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