Subdirectly irreducibles in some varieties of algebras having the semilattice structure (Q1200944)

With a type \(\tau: F\to\omega\) \((F\neq\emptyset)\) and a symbol \(\lor\not\in F\) we associate a new type \(\tau^*: F\cup\{\lor\}\to\omega\), where \(\tau^*(\lor)=2\) and \(\tau^*| F=\tau\). A unary term \(q(x)\) of type \(\tau^*\) in which the outermost symbol belongs to \(F\) is called \(F\)-external. For a fixed \(F\)-external unary term \(q(x)\) of type \(\tau^*\) denote by \(S_ 0\) the variety determined by the following identities: (1) \(x\lor x=x\); (2) \(x\lor y=y\lor x\); (3) \((x\lor y)\lor z=x\lor (y\lor z)\); (4) \(x\lor q(x)=x\); (5) \(q(q(x))=q(x)\); (6) \(q(x\lor y)=q(x)\lor q(y)\); (7) \(q(f(x_ 0,\dots,x_{n-1}))=f(q(x_ 0),\dots,q(x_{n-1}))=f(x_ 0,\dots,x_{n-1})\) for \(f\in F\), \(\tau^*(f)=n\). In the paper we describe all subdirectly irreducible algebras in varieties \(S_ 0\lor K\) for varieties \(K\) of type \(\tau^*\) satisfying identities (1)--(3) and the identity \(q(x)=x\).