Intervals in \(l\)-groups as \(L\)-algebras
\(L\)-algebras have been introduced by \textit{W. Rump} in [J. Algebra 320, No. 6, 2328--2348 (2008; Zbl 1158.06009)] as algebras \((X, \rightarrow, 1)\) of type \((2, 1)\) satisfying the following conditions: \((L_1)\) \(x\rightarrow x=x\rightarrow 1=1\), \(1\rightarrow x=x,\) \((L_2)\) \((x\rightarrow y)\rightarrow (x\rightarrow z)=(y\rightarrow x)\rightarrow (y\rightarrow z),\) \((L_3)\) \(x\rightarrow y=y\rightarrow x=1 \Rightarrow x=y.\) An \(L\)-algebra \(X\) is said to be semiregular if it satisfies the condition \((x\rightarrow y)\rightarrow z)\rightarrow ((y\rightarrow x)\rightarrow z)= ((x\rightarrow y)\rightarrow z)\rightarrow z\) for all \(x, y, z\in X\). In [loc. cit.], the first author employed a self-similar closure \(S(X)\) of an \(L\)-algebra \(X\) to derive a criterion for \(X\) to be representable as an interval in a lattice-ordered group. The purpose of the present paper is to improve this criterion without using the embeding. It is proved that a semiregular \(L\)-algebra with negation is a left hoop. A semiregular \(L\)-algebra is characterized as a left hoop satisfying some special conditions. Finally, the authors prove that an \(L\)-algebra is representable as an interval in a lattice-ordered group if and only if it is semiregular with a smallest element and bijective negation. I consider that the reviewed article contains valuable results and can serve as a starting point for other studies on this subject.
scientific article; zbMATH DE number 6029769
scientific article