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 \(\mathbf{DRL}(m)\cap\mathbf{DRL}(n)=\mathbf{DRL}(\gcd\{m,n\})\).


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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