Varieties of semiassociative relation algebras and tense algebras (Q2310438)

This article is concerned with join-irreducible covers of two varieties within five lattices of subvarieties. The varieties are concerned with tense algebras and with variants of the relation algebras of Tarski. The two varieties are the variety \(\mathrm{var}(A_3)\) generated by \(A_3\), which is the minimal subalgebra of the full relation algebra on a 3-element set, and the variety \(\mathrm{var}(T_0)\) generated by \(T_0\), which is the 2-element Boolean algebra with the identity function added twice. A \textit{tense algebra} is a Boolean algebra \(A\) with two additional one-place functions \(f,g\) satisfying the inequalities \[ f(x\wedge-g(y))\leq f(x)\wedge-y\quad\hbox{and}\quad g(y\wedge-f(x))\leq g(y)\wedge-x. \] The lattice of subvarieties of the variety of tense algebras is denoted by \(\Lambda_{TA}\). \(T_0\) is a tense algebra. A tense algebra \(A\) is \textit{total} \(\iff\) for all \(x\not=0\), \(f(x)+g(x)=1\). The lattice of subvarieties of the variety generated by the total tense algebras is denoted by \(\Lambda_{TTA}\). The variants of Tarski's relation algebras are as follows. The precise definitions are somewhat complicated, and we omit them. \(\Lambda_{NA}\), \(\Lambda_{SA}\), and \(\Lambda_{RSA}\) are the lattice of subvarieties of the lattice of non-associative relation algebras, the lattice of semiassociative relation algebras, and the lattice of reflexive subadditive symmetric semiassociative relation algebras, respectively. Now the two main results of the paper are: (1) \(\mathrm{var}(T_0)\) has \(2^{\aleph_0}\) join-irreducible covers in \(\Lambda_{TTA}\) and \(\Lambda_{TA}\). (2) \(\mathrm{var}(A_3)\) has \(2^{\aleph_0}\) covers in \(\Lambda_{RSA}\), \(\Lambda_{SA}\) and \(\Lambda_{NA}\).