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Indecomposable, projective and flat Sposets
 Comm. Algebra
"... Abstract. For a monoid S, a (left) Sact is a nonempty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Right Sacts A can also be defined, and a tensor product A ⊗S B (a set) can be defined that has the customary universal ..."
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Abstract. For a monoid S, a (left) Sact is a nonempty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Right Sacts A can also be defined, and a tensor product A ⊗S B (a set) can be defined that has the customary universal property with respect to balanced maps from A×B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (Walter de Gruyter, New York, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets (Sposets). The present paper is devoted to such a generalization. A unique decomposition theorem for Sposets is given, based on strongly convex, indecomposable Ssubposets, and a structure theorem for projective Sposets is given. A criterion for when two elements of the tensor product of Sposets is given, which is then applied to investigate several flatness properties. 1. introduction and preliminaries If S is a monoid, it is wellknown that Sacts (also called Ssets, Ssystems, Sautomata, etc.) play an important role not only in studying properties of monoids, but also in other mathematical areas, such as graph theory and algebraic automata theory (see [1], [2]). Flatness and projectivity are important topics in the study of acts over monoids. Flat acts are important in studying amalgamation properties of monoids (see [1] and references contained therein), and projective and indecomposable acts play an essential role in studying the
ALGEBRA IN SUPEREXTENSIONS OF GROUPS, IV: REPRESENTATION THEORY
, 811
"... Abstract. Given a group X we study the algebraic structure of the compact righttopological semigroup λ(X) consisting of maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in t ..."
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Abstract. Given a group X we study the algebraic structure of the compact righttopological semigroup λ(X) consisting of maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup P(X) P(X) of all selfmaps of the powerset P(X) and show that the image of λ(X) in P(X) P(X) coincides with the semigroup λ(X, P(X)) of all functions f: P(X) → P(X) that are equivariant, monotone and symmetric in the sense that f(X \ A) = X \ f(A) for all A ⊂ X. Using this representation we describe the minimal ideal K(λ(X)) and minimal left ideals of the superextension λ(X) of each somewhat commutative group X, where a group X is somewhat commutative if the product xy of any two elements x, y ∈ X belongs to the semigroups generated by the set {yx, yx−1, y−1x, y−1x−1}. We prove that for each commutative group X (admitting no epimorphism onto the quasicyclic group C2∞) each minimal left ideal of the supserextension λ(X) is alegebraically (and topologically) isomorphic to the product
Abstract DRAFT
"... Build your own probability monads Probability is often counterintuitive, and it always involves a great deal of math. This is unfortunate, because many applications in robotics and AI increasingly rely on probability theory. We introduce a modular toolkit for constructing probability monads, and sh ..."
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Build your own probability monads Probability is often counterintuitive, and it always involves a great deal of math. This is unfortunate, because many applications in robotics and AI increasingly rely on probability theory. We introduce a modular toolkit for constructing probability monads, and show that it can be used for everything from discrete distributions to weighted particle filtering. This modular approach allows us to present a single, easytouse API for working with many kinds of probability distributions. Our toolkit combines several existing components (the list monad, the Rand monad, and the MaybeT monad transformer), with a stripped down version of WriterT Prob, and a new monad for sequential Monte Carlo sampling. Using these components, we show that MaybeT can be used to implement Bayes ’ theorem. We also show how to implement a monad for weighted particle filtering.
Denoting by Aut(M) the automorphism group of M∈Iso(A), it is easy to see that
"... Abstract. Let A be a finitely generated semigroup with 0. An A–module overF1 (also called an A–set), is a pointed set (M,∗) together with an action of A. We define and study the Hall algebraHA of the categoryCA of finite A–modules.HA is shown to be the universal enveloping algebra of a Lie algebranA ..."
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Abstract. Let A be a finitely generated semigroup with 0. An A–module overF1 (also called an A–set), is a pointed set (M,∗) together with an action of A. We define and study the Hall algebraHA of the categoryCA of finite A–modules.HA is shown to be the universal enveloping algebra of a Lie algebranA, called the Hall Lie algebra ofCA. In the case of the〈t 〉 the free monoid on one generator〈t〉, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent〈t〉modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when A is a quotient of〈t 〉 by a congruence, and the monoid G∪{0} for a finite group G. 1. introduction The aim of this paper is to define and study the Hall algebra of the category of settheoretic representations of a semigroup. Classically, Hall algebras have been studied in the context abelian categories linear over finite fieldsFq. Given such a categoryA, finitary in the sense that Hom(M, N) and Ext 1 (M, N) are finitedimensional ∀ M, N∈A (and therefore finite sets), we may construct fromAan associative algebraHA defined overZ 1, called the RingelHall algebra ofA. As aZ–module,HA is freely generated by the isomorphism classes of objects inA, and its structure constants are expressed in terms of the number of extensions between objects. Explicitly, if M and N denote two isomorphism classes, their product inHA is given by