Minimal characteristic algebras for some properties of identities (Q699918)

A structural property of an identity of type \(\tau\) is called hereditary if for every set \(I\) of identities of type \(\tau\) having the property \(p\) every consequence of \(I\) has the property \(p\) too. An algebra \(A\) of type \(\tau\) is called characteristic for a hereditary property \(p\) if, for every variety \(\mathcal V\) of type \(\tau\), \(A\in\mathcal V\) if and only if every identity from \(\text{Id}(\mathcal V)\) has the property \(p\). In the article under review the author consider minimal characteristic algebras (that is, characteristic algebras of the minimal possible cardinality) for several hereditary properties, namely: (a) ``to be externaly compatible'' (an identity \(u=v\) is called externally compatible if either \(u\) and \(v\) are the same variable or most of the external fundamental operation symbols in \(u\) and \(v\) are identical); (b) ``to be outermost'' (an identity \(u=v\) is called left [right] outermost if \(u\) and \(v\) have the same first [last] variables; an identity is called outermost if it is both left outermost and right outermost); (c) ``to be normal and regular''; (d) ``to be regular and left outermost''; (e) ``to be normal and left outermost''; (f) ``to be regular and outermost''.