245 research outputs found
Classification of grouplike categories
In this paper we study grouplike monoids, these are monoids that contain a
group to which we add an ordered set of idempotents. We classify finite
categories with two objects having grouplike endomorphism monoids, and we give
a count of certain categories with grouplike monoids.Comment: Minor changes in Lemma 3.18. Added Definition 4.3, Lemma 4.4 and
Remark 4.5. And the proof of Proposition 4.6 is improve
Bianchi spaces and their 3-dimensional isometries as S-expansions of 2-dimensional isometries
In this paper we show that some 3-dimensional isometry algebras, specifically
those of type I, II, III and V (according Bianchi's classification), can be
obtained as expansions of the isometries in 2 dimensions. It is shown that in
general more than one semigroup will lead to the same result. It is impossible
to obtain the algebras of type IV, VI-IX as an expansion from the isometry
algebras in 2 dimensions. This means that the first set of algebras has
properties that can be obtained from isometries in 2 dimensions while the
second set has properties that are in some sense intrinsic in 3 dimensions. All
the results are checked with computer programs. This procedure can be
generalized to higher dimensions, which could be useful for diverse physical
applications.Comment: 23 pages, one of the authors is new, title corrected, finite
semigroup programming is added, the semigroup construction procedure is
checked by computer programs, references to semigroup programming are added,
last section is extended, appendix added, discussion of all the types of
Bianchi spaces is include
Monoids of modules and arithmetic of direct-sum decompositions
Let be a (possibly noncommutative) ring and let be a class
of finitely generated (right) -modules which is closed under finite direct
sums, direct summands, and isomorphisms. Then the set
of isomorphism classes of modules is a commutative semigroup with operation
induced by the direct sum. This semigroup encodes all possible information
about direct sum decompositions of modules in . If the endomorphism
ring of each module in is semilocal, then is a Krull monoid. Although this fact was observed nearly a decade ago, the
focus of study thus far has been on ring- and module-theoretic conditions
enforcing that is Krull. If
is Krull, its arithmetic depends only on the class group of and the set of classes containing prime divisors. In this paper
we provide the first systematic treatment to study the direct-sum
decompositions of modules using methods from Factorization Theory of Krull
monoids. We do this when is the class of finitely generated
torsion-free modules over certain one- and two-dimensional commutative
Noetherian local rings.Comment: Pacific Journal of Mathematics, to appea
The number of nilpotent semigroups of degree 3
A semigroup is \emph{nilpotent} of degree 3 if it has a zero, every product
of 3 elements equals the zero, and some product of 2 elements is non-zero. It
is part of the folklore of semigroup theory that almost all finite semigroups
are nilpotent of degree 3.
We give formulae for the number of nilpotent semigroups of degree 3 with
elements up to equality, isomorphism, and isomorphism or
anti-isomorphism. Likewise, we give formulae for the number of nilpotent
commutative semigroups with elements up to equality and up to isomorphism
Teaching Time Savers: The Exam Practically Wrote Itself!
When I first started teaching, creating an exam for my upper division courses was a genuinely exciting process. The material felt fresh and relatively unexplored (at least by me), and I remember often feeling pleasantly overwhelmed with what seemed like a vast supply of intriguing and engrossing exam-ready problems. Crafting the perfect exam, one that was noticeably inviting, exceedingly fair, and unavoidably illuminating, was a real joy
Fighting bit Rot with Types (Experience Report: Scala Collections)
We report on our experiences in redesigning Scala\u27s collection
libraries, focussing on the role that type systems play in keeping
software architectures coherent over time. Type systems can make
software architecture more explicit but, if they are too weak, can
also cause code duplication. We show that code duplication can be
avoided using two of Scala\u27s type constructions: higher-kinded types
and implicit parameters and conversions
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