Adjunctions in Monoids
نویسنده
چکیده
be morphisms of monoids, considered as functors. Let the functor f is left adjoint to the functor g. Is it true then that f (or, what is the same, g) is always an isomorphism? In [1], p.136, this question was posed as an open question. Here I answer this question and the answer is no. To prove this, I will construct a Birkhoff variety of algebras, which is naturally equivalent to the category of adjunctions in monoids, and consider its initial object which is a monoid generated by 2 × N free variables subject to a certain set of relations. An application of M. H. A. Newman’s reduction theorem ([4], cited by [3]) permits one to describe the canonical form of elements in the monoid and, in particular, to negatively answer the question posed. Let φ:N −→ M, (2)
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