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On directed groups. (English. Russian original) Zbl 0714.06007
Sib. Math. J. 30, No. 6, 895-902 (1989); translation from Sib. Mat. Zh. 30, No. 6(178), 78-86 (1989).
A directed set (M;\(\leq)\) can be transformed into an algebra (M;\(\sigma\)) of signature \(\sigma =\{\vee,\wedge \}\) which contains all binary operations \(\vee\) and \(\wedge\) defined in the following way: if x,y\(\in M\), \(x\leq y\) we set \(x\vee y=y\) and \(x\wedge y=x\). If x and y are uncomparable, we choose (and fix) for \(x\vee y\) an upper bound of x and y. Dually for \(x\wedge y\). The operations can be defined as commutative ones. 1.2 Theorem. Let (M;\(\leq)\) be a directed set and (M;\(\sigma\)) an algebra of signature \(\sigma =\{\vee,\wedge \}\) which is constructed by means of \(\leq\) in the described way. Then there holds in (M;\(\sigma\)) H1) \(x\vee x=x\), \(x\wedge x=x\), H2) \(x\vee y=y\vee x\), \(x\wedge y=y\wedge x\), H3) \(x\vee (x\vee y)=x\vee y\), \(x\wedge (x\wedge y)=x\wedge y\), H4) \((x\vee y)\vee z=x\vee ((x\vee y)\vee z)\) and dually, H5) \(x\vee (x\wedge y)=x\) and dually. Conversely, if an order relation \(\leq\) is introduced in an algebra (M;\(\sigma\)) so that \(x\leq y\) iff \(x\vee y=y\) (or \(x\wedge y=x)\) then (M;\(\leq)\) is a directed set. If \((G;\cdot,e,^{-1})\) is a group and (G;\(\leq)\) a directed set, then \((G;\cdot,e,^{-1},\leq)\) is called a directed group if \(x\leq y\) implies zxt\(\leq zyt\) for arbitrary z,t\(\in G\). 2.2 Theorem. If \((G;\cdot,e,^{-1},\leq)\) is a directed group then there can be defined operations \(\vee\) and \(\wedge\) in G so that H1- H5 hold and moreover H6) \(zxt\vee z(x\vee y)t=z(x\vee y)t\) and dually, and H7) \((x\vee y)^{-1}=x^{-1}\vee y^{-1}\) and dually. Conversely, every algebra \((G;\cdot,e,^{-1},\vee,\wedge)\) so that \((G;\cdot,e,^{- 1})\) is a group which fulfils H1-H7, turns out to be a directed group if we put \(x\leq y\) iff \(x\vee y=y\). Other properties of directed groups are deduced which are near to the \(\ell\)-group properties.
Reviewer: F.Šik

MSC:
06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
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References:
[1] L. Fuks, Partially Ordered Algebraic Systems [in Russian], Nauka, Moscow (1965).
[2] A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).
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