Cayley Posets
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Cayley Posets Ignacio Garc´ıa-Marco , Kolja Knauer and Guillaume Mercui-Voyant Abstract. We introduce Cayley posets as posets arising naturally from pairs S < T of semigroups, much in the same way that a Cayley graph arises from a (semi)group and a subset. We show that Cayley posets are a common generalization of several known classes of posets, e.g., posets of numerical semigroups (with torsion) and more generally affine semigroups. Furthermore, we give Sabidussi-type characterizations for Cayley posets and for several subclasses in terms of their endomorphism monoid. We show that large classes of posets are Cayley posets, e.g., series–parallel posets and (generalizations of) join-semilattices, but also provide examples of posets which cannot be represented this way. Finally, we characterize (locally finite) auto-equivalent posets (with a finite number of atoms)—a class that generalizes a recently introduced notion for affine semigroups—as those posets coming from a finitely generated submonoid of an abelian group. Mathematics Subject Classification. 06A11, 06A07, 20M99. Keywords. Numerical semigroup, Cayley poset, Monoid, Semigroup.
1. Introduction Cayley graphs of groups are a classical topic in algebraic graph theory. They play a prominent role in (books devoted to) the area, see, e.g., [10]. A particular and central result of the theory, due to Sabidussi [44], characterizes Cayley graphs of groups via the action of their automorphism group. Cayley graphs of monoids and semigroups have been less studied, but still there is a considerable amount of work, see, e.g., [30] for a book and the references therein. In semigroups, analogues of the above result of Sabidussi remain open. Characterizations of Cayley graphs of certain classes of semigroups have been subjected to some research effort, see [1,11–13,18,20–24,26,32,35,36,51,52]. Conversely, also characterizations of semigroups admitting Cayley graphs with certain (mostly topological) properties have been investigated extensively [15,17,19,28,29,40,45–47,54]. With respect to groups, there is a wellknown book concerning the topic [53]. 0123456789().: V,-vol
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I. Garc´ıa-Marco
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N abcd a b c d
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S abcd a b c d
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Figure 1. The N-poset is a monoid poset via P (N, {a, c, d}) and is full as witnessed by P (S, S) In the present paper, we pursue questions of this type in a setting which naturally excludes groups from the picture. Namely, we study semigroups whose Cayley graph yields a partially ordered set (poset). We call the resulting posets Cayley posets. Such objects arise also naturally from considering (relative) Green’s relations on semigroups. An important class of Cayley posets are numerical semigroups (see the books [2,41,43]), and more generally numerical semigroups with torsion (see [9,31]) and affine semigroups (see the books [6,37,42,49]). Further examples include families defined to construct diamond free posets in [7]. Let us define the setting of this article in the slightly more general l
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