Semilattice modes. I: The associated semiring.(English)Zbl 0848.08005

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.

MSC:

 08B99 Varieties 16Y60 Semirings
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References:

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