Abstract
AbstractFinitary monads on Pos are characterized as precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analogously, finitary enriched monads on Pos are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.
Highlights
Algebraic specifications of data types are often given in terms of operations and equations
We have considered the special case of signatures where each poset is discrete, i.e. we just have a set of operation symbols of arity ; for emphasis, we will call such signatures discrete. (N.B.: This terminology differs from the way the word discrete is used in the concept of discrete Lawvere theory (Power, 2005) where it refers to the arities of operations rather than the objects .)
We have investigated the analogous situation for the category of posets
Summary
Algebraic specifications of data types are often given in terms of operations and equations. The first author, Dostál, and Velebil (2021) proved that for every such variety V the free-algebra monad TV is enriched and strongly finitary in the sense of Kelly and Lack (1993) This means that the functor TV is the left Kan extension of its restriction along the full embedding E : Posfd → Pos of finite discrete posets: TV = LanE (TV · E). Every strongly finitary monad on Pos is isomorphic to the free-algebra monad of a variety in this classical sense This answers our question above affirmatively: arities in Posf are necessary if all (possibly enriched) finitary monads are to be characterized via inequations. Op. cit. describes a sound and complete sequent system for inequational reasoning, which yields an alternative description of the free-algebra monad of an inequational theory
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