122 research outputs found
Profinite lambda-terms and parametricity
Combining ideas coming from Stone duality and Reynolds parametricity, we
formulate in a clean and principled way a notion of profinite lambda-term
which, we show, generalizes at every type the traditional notion of profinite
word coming from automata theory. We start by defining the Stone space of
profinite lambda-terms as a projective limit of finite sets of usual
lambda-terms, considered modulo a notion of equivalence based on the finite
standard model. One main contribution of the paper is to establish that,
somewhat surprisingly, the resulting notion of profinite lambda-term coming
from Stone duality lives in perfect harmony with the principles of Reynolds
parametricity. In addition, we show that the notion of profinite lambda-term is
compositional by constructing a cartesian closed category of profinite
lambda-terms, and we establish that the embedding from lambda-terms modulo
beta-eta-conversion to profinite lambda-terms is faithful using Statman's
finite completeness theorem. Finally, we prove that the traditional Church
encoding of finite words into lambda-terms can be extended to profinite words,
and leads to a homeomorphism between the space of profinite words and the space
of profinite lambda-terms of the corresponding Church type
Boolean like algebras
Using Vaggione’s concept of central element in a double pointed algebra, we introduce the notion of Boolean like variety as a generalization of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove
that a double pointed variety is discriminator i↵ it is semi-Boolean like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterization of double pointed discriminator varieties. Moreover, we show that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional regular operations
On Noncommutative Generalisations of Boolean Algebras
Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions
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