Semilattice modes. I: The associated semiring
A mode \({\mathbf A}\) is an idempotent, entropic algebra, so any basic operation \(f\) of \({\mathbf A}\) satisfies \(f(x,...,x) = x\) and any two term operations of arities \( m\) and \(n\) satisfy \[ f ( g(u_{11}, \dots, u_{1n}), \dots, g (u_{m1}, \dots, u_{mn})) = g (f(u_{11}, \dots, u_{m1}), \dots, f (u_{1n}, \dots, u_{mn})), \] where \([u_{ij}]\) is an \(m \times n\) matrix of elements of \(A\). Clearly, since modes are equationally defined, any mode generates a variety of modes. This paper considers semilattice modes which are modes with a binary term satisfying the conditions for a semilattice and shows that each variety of semilattice modes can be associated with a semilattice ring. For these purposes, a semiring is an algebra \(R = \langle R,+, \cdot, 0,1 \rangle\) of type \(\langle 2,2,0,0 \rangle\) such that \(\langle R,+,0 \rangle\) is a commutative monoid, \(\langle R, \cdot, 1,0 \rangle\) is a monoid with zero and \(\cdot\) both left and right distributes over \(+\). A semimodule over a semiring is defined in the obvious way. The semirings of interest here satisfy the additional law \(1 + r = 1\). It is easy to verify that in this situation \(+\) is idempotent, and so both \(\langle R, + \rangle\) and any semimodule over \(R\) are semilattices. Let \(F_{\mathcal V} (x,y)\) be the free algebra of rank two in a variety \({\mathcal V}\) of semilattice modes, and let \(R\) be the subset of \(F_{\mathcal V} (x,y)\) consisting of those terms \(t\) satisfying \(t + y = y\). For any \(t \in F_{\mathcal V} (x,y)\), \(e_t\) denotes the endomorphism of \(F_{\mathcal V} (x,y)\) determined by \(x \mapsto t\), \(y \mapsto y\). For any \(s,t \in F_{\mathcal V} (x,y)\), \(s \circ t\) denotes \(e_t (s)\). Then the algebra \(R ({\mathcal V}) = \langle R,+, \circ, x + y, y \rangle\) is a commutative semiring satisfying \(1 + r = 1\). Moreover, the subdirectly irreducible elements of \({\mathcal V}\) may be interpreted as semimodules over \(R({\mathcal V})\). In this paper it is shown how various properties of the variety are reflected in those of the corresponding semiring. In particular, this relationship is used to prove that a variety of semilattice modes has the congruence extension property.
scientific article; zbMATH DE number 819276