## 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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