Distributive elements of the lattice of semigroup varieties
Let \textbf{SEM} be the lattice of all semigroup varieties. A pair of identities \(wx=xw=w\), where the letter \(x\) does not occur in the word \(w\), is written as \(w=0\). Let \({\mathcal{D}}\) be the set of the following varieties of semigroups: \({\mathcal{SEM}}\) = variety of all semigroups; \({\mathcal{SL}}=\text{var}\{x=x^2, xy=yx\}\); \({\mathcal{M}}_1=\text{var}\{x^2y=xyx=yx^2=0\}\); \({\mathcal{M}}^n_1=\text{var}\{x^2y=xyx=yx^2=x_1x_2\dots x_n=0\}\); \({\mathcal{M}}_2=\text{var}\{x^2=xyx=0\}\); \({\mathcal{M}}^n_2=\text{var}\{x^2=xyx=x_1x_2\dots x_n=0\}\) \((n\in \mathbb{N})\). The authors prove that the set of all distributive elements of the lattice \textbf{SEM} is equal to the set \({\mathcal D}\cup\{{\mathcal M}_1\vee {\mathcal S}{\mathcal L}, \;{\mathcal M}^n_1\vee {\mathcal S}{\mathcal L}, \;{\mathcal M}_2\vee {\mathcal S}{\mathcal L}, \;{\mathcal M}^n_2\vee {\mathcal S}{\mathcal L}\}\).
scientific article; zbMATH DE number 5840717
scientific article