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To binary trees, two-ary integers are what usual integers are to natural numbers, seen as unary trees. We can represent two-ary integers as binary trees too, yet with leaves labelled by binary words and with a structural restriction. In a sense, they are simpler than the binary trees, they relativize. Hence, contrary to the extensions known from Arithmetic and Algebra, this integer extension does not make the starting objects more complex. We use a semantic construction to get this extension. This...
It is shown that every concretizable category can be fully embedded into the category of accessible set functors and natural transformations.
A -labeled -poset is an (at most) countable set, labeled in the set , equipped with partial orders. The collection of all -labeled -posets is naturally equipped with binary product operations and -ary product operations. Moreover, the -ary product operations give rise to
A Σ-labeled n-poset is an (at most) countable set,
labeled in the set Σ, equipped with n partial orders.
The collection of all Σ-labeled n-posets is naturally
equipped with n binary product operations and
nω-ary product operations.
Moreover, the ω-ary product operations
give rise to nω-power operations.
We show that those Σ-labeled n-posets that can be generated from
the singletons by the binary and ω-ary
product operations form the free algebra on Σ
in a variety axiomatizable by an infinite collection...
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