Quantifiers on m-semilattices
by
Jānis Cīrulis
University of Latvia, Riga, Latvia

A multiplicative semilatice, or an m-semilattice, is an algebra (A, ∨, ·, 0), where (A, ∨, 0) is a semilattice with the least element 0, (A, ·, 0) is a groupoid with the multiplicative zero 0, and the multiplication · is left and right distributive over ∨. We are interested mainly in integral m-semilattices. A (left) quantifier on A is defined to be a selfmap ∇ of A satisfying the conditions

∇(a ∨b) = ∇a ∨∇b,  ∇0 = 0,  a ≤ ∇a,  ∇(a ·∇b) = ∇a ·∇b.

The topic of the talk is various characterisations of quantifiers on abstract and functional m-semilattices. In particular, it is shown that there is a one-to-one correspondence between A-valued equivalence relations on some set X and a certain subset of quantifiers on the functional m-semilattice A^X.

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Date received: May 13, 2007