Abstract.
It is shown that the category of directed graphs is isomorphic to a subcategory of the variety S of all pseudocomplemented semilattices which contains all homomorphisms whose images do not lie in the subvariety B of all Boolean pseudocomplemented semilattices. Moreover, the functor exhibiting the isomorphism may be chosen such that each finite directed graph is assigned a finite pseudocomplemented semilattice. That is to say, it is shown that the variety S of all pseudocomplemented semilattices is finite-to-finite B-relatively universal.
This illustrates the complexity of the endomorphism monoids of pseudocomplemented semilattices since it follows immediately that, for any monoid M, there exists a proper class of non-isomorphic pseudocomplemented semilattices such that, for each member S, the endomorphisms of S which do not have an image contained in the skeleton of S form a submonoid of the endomorphism monoid of S which is isomorphic to M.
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To Věra Trnková on the occasion of her 70th birthday.
Received June 17, 2006; accepted in final form May 8, 2007.
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Adams, M.E., Schmid, J. Pseudocomplemented semilattices are finite-to-finite relatively universal. Algebra univers. 58, 303–333 (2008). https://doi.org/10.1007/s00012-008-2071-3
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DOI: https://doi.org/10.1007/s00012-008-2071-3