## Varieties of distributive rotational lattices.(English)Zbl 1318.06005

A rotational lattice is a (universal) algebra $$\mathbf L=(L;\vee,\wedge,g)$$, where $$(L;\vee,\wedge)$$ is a lattice, $$g$$ is an automorphism of this lattice and $$g^n(x)=g(g(\cdots g(x)\cdots))=x$$ for all $$x\in L$$ (with $$n$$ copies of $$g$$). The class of these algebras, for a given $$n$$, is denoted by $$\mathbf{RL}(n)$$. If the lattice $$(L;\vee,\wedge)$$ is distributive, then $$\mathbf L$$ is called distributive rotational lattice and $$\mathbf{DRL}(n)$$ means the class of all distributive members of $$\mathbf{RL}(n)$$. It can be shown that $$\mathbf{DRL}(n)$$ are varieties for any natural $$n$$. Note that the class of all distributive rotational lattices is not a variety.
The authors investigate the varieties $$\mathbf{DRL}(n)$$ and characterize the subdirectly irreducible members of these classes. More precisely, let $$B_n=(B_n;\vee,\wedge)$$ denote the Boolean lattice of length $$n$$. Let $$a^{(n)}_0,\ldots,a^{(n)}_{n-1}$$ be the atoms of $$B_n$$. Define the automorphism $$g$$ of $$B_n$$ as follows: $$g(a^{(n)}_i)=a^{(n)}_{i+1}$$ where $$i+1$$ is modulo $$n$$. This way we obtain $$\mathbf B_n=(B_n;\vee,\wedge,g)\in\mathbf{DRL}(n)$$ where $$n$$ is minimal with the property $$g^n(x)=x$$ for each $$x\in\mathbf B_n$$.
Main results: (1) The subdirectly irreducible distributive rotational lattices are exactly the algebras $$\mathbf B_n$$, $$n\in N$$. In addition, the algebras $$\mathbf B_n$$ are simple. (2) $$\mathbf B_m\in\mathbf{DRL}(n)$$ iff $$m$$ divides $$n$$. (3) $$\mathbf{DRL}(m)\vee\mathbf{DRL}(n)=\mathbf{DRL}(\text{lcm}\{m,n\})$$ and $$\mathbf{DRL}(m)\cap\mathbf{DRL}(n)=\mathbf{DRL}(\gcd\{m,n\})$$.

### MSC:

 06B75 Generalizations of lattices 06B20 Varieties of lattices 06D05 Structure and representation theory of distributive lattices 08B26 Subdirect products and subdirect irreducibility
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