Download PDF by Salzmann H.: 4-Dimensional projective planes of Lenz type III

By Salzmann H.

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Finally, via the q-theory, continuous self-maps of PV are introduced, leading to algebraic and continuous lattices (in the sense of [97]), topological semigroups (under composition) and various types of quantales (complete lattices with associative multiplications) [303]. “Coordinate systems” for continuous operators will be pseudovarieties of relational morphisms, created by B. Tilson and the second author (with some influence of the first author), which we shall proceed to describe shortly. Also questions of decidability and undecidability for all the above can be considered; see [2, 36, 281, 285].

In this way, relational morphisms are “adding arrows” to the category FSgp by “turning around surmorphisms” (a homotopy-theoretic idea [257]). The category (FSgp,RM) is then defined as the category with objects finite semigroups and with arrows from S to T consisting of relational morphisms, that is, Hom(S, T ) = {ϕ : S → T | ϕ is a relational morphism}. The initial object is still ∅ and the terminal object is still {1}. Note that ∅ → T is the unique arrow from ∅ to T . The existence of an arrow S → ∅ implies S = ∅.

16) The resulting semigroup M N is in fact a monoid with identity (0M , 1N ). The projection M N N is a surjective homomorphism of monoids. 1. 15) is an associative product and (0M , 1N ) is the identity for M N . We are deliberately avoiding usage of the symbol ∗ for the semidirect product of two semigroups (as is frequently done in the literature [7, 27, 85, 364]) to avoid confusion with the free product of monoids and also because ∗ is too symmetric a symbol for a non-symmetric product. 30).

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4-Dimensional projective planes of Lenz type III by Salzmann H.

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