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1.
In this paper we investigate under which conditions a monoid R is defined by the endomorphism monoid of an act over R. More
precisely, we ask when an isomorphism between two such endomorphism monoids over monoids R1 and R2 is induced by a semilinear isomorphism. The question is considered also for ordered and for topological monoids. On the way
we characterize monoids over which all projective acts are free. An abstract of this paper appeared in the Proceedings of
the Conference on Semigroups, Szeged 1972. 相似文献
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Communicated by Boris M. Schein 相似文献
4.
We prove, that, given a finite graph Y there exists a finite monoid (semigroup with unity) M such that any graph X whose endomorphism monoid is isomorphic to M contains a subdivision of Y. This contrasts with several known results on the simultaneous prescribability of the endomorphism monoid and various graph theoretical properties of a graph. It is also related to the analogous problems on graphs having a given permutation group as a restriction of their automorphism group to an invariant subset. 相似文献
5.
In 1971, Stenström published one of the first papers devoted to the problem of when, for a monoid S and a right S -act A S , the functor A? (from the category of left acts over S into the category of sets) has certain limit preservation properties. Attention at first focused on when this functor preserves pullbacks and equalizers but, since that time, a large number of related articles have appeared, most having to do with when this functor preserves monomorphisms of various kinds. All of these properties are often referred to as flatness properties of acts . Surprisingly, little attention has so far been paid to the obvious questions of when A S ? preserves all limits, all finite limits, all products, or all finite products. The present article addresses these matters. 相似文献
6.
A concrete category is almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. This paper characterizes almost universal varieties of commutative semigroups. As a consequence we obtain that for every infinite cardinal κ there exists a commutative semigroup of cardinality κ such that it has exactly two endomorphisms, the identity endomorphism and a single constant endomorphism. 相似文献
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A monoidM and a latticeL arealgebraic if there is an algebraA with endomorphism monoid EndA M and subalgebra lattice SuA L. For each chainC we characterize those monoidsM for whichM and C are algebraic. In particular we show that a finite monoidM is algebraic with the three-chain iff the equalizers ofM form a chainE 3. The same assertion however fails for infinite monoids. This generalizes the corresponding result for two-chains and solves a problem posed by B. Jónsson ([2], p. 147). We settle the same question for all longer chainsK.
Presented by Ivo Rosenberg. 相似文献
9.
Projective acts whose endomorphism monoids are left or right (semi-) hereditary are characterized. For example, it is shown
that for a noncyclic free or projective S-act P, End P is left (semi) hereditary if and only if P ≈ Se1 Π Se2 and e1Se1, e2Se2 are groups. 相似文献
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In 1981 Edgar Enochs conjectured that every module over a unitary ring has a flat cover. He finally proved this conjecture in 2001, in a paper that included an independent proof by Bican and El Bashir. Enochs had in fact considered different types of covers as early as 1963, for example injective and torsion free covers, and since then a great deal of effort has been spent on their study. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the ‘other’ type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective covers and demonstrate that in some cases the results are quite different from the module case. 相似文献
12.
Ulrich Knauer 《Semigroup Forum》1971,3(1):359-370
In this paper we shall consider a non-additive category of A-modules, that is, instead of a ring A we take a monoid A which
acts on sets from the left. These objects will be called A-acts. We investigate indecomposable A-acts and generators and characterize
projectives in this category. For a given monoid A we describe all monoids B such that the category of B-acts is equivalent
to the category of A-acts. In particular we find that equivalence of these categories yields an isomorphism between the monoids
A and B if A is a group or finite or commutative. This differs from the additive case where the categories of modules over
a commutative field and its ring of nxn matrices are equivalent. Finally we give examples of non-isomorphic monoids A and
B such that the corresponding categories are equivalent. 相似文献
13.
Ronald E. Prather 《Linear algebra and its applications》1976,13(3):201-205
A matrix characterization is obtained for the epimorphisms in the category of finitely generated free monoids. It follows from our result that it is effectively decidable whether a given morphism is an epimorphism. The corresponding question for monomorphisms has been answered in the algorithm of Sardinas and Patterson. 相似文献
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On covers of cyclic acts over monoids 总被引:1,自引:0,他引:1
In (Bull. Lond. Math. Soc. 33:385–390, 2001) Bican, Bashir and Enochs finally solved a long standing conjecture in module theory that all modules over a unitary ring
have a flat cover. The only substantial work on covers of acts over monoids seems to be that of Isbell (Semigroup Forum 2:95–118,
1971), Fountain (Proc. Edinb. Math. Soc. (2) 20:87–93, 1976) and Kilp (Semigroup Forum 53:225–229, 1996) who only consider projective covers. To our knowledge the situation for flat covers of acts has not been addressed and this
paper is an attempt to initiate such a study. We consider almost exclusively covers of cyclic acts and restrict our attention
to strongly flat and condition (P) covers. We give a necessary and sufficient condition for the existence of such covers and for a monoid to have the property that
all its cyclic right acts have a strongly flat cover (resp. (P)-cover). We give numerous classes of monoids that satisfy these conditions and we also show that there are monoids that do
not satisfy this condition in the strongly flat case. We give a new necessary and sufficient condition for a cyclic act to
have a projective cover and provide a new proof of one of Isbell’s classic results concerning projective covers. We show also
that condition (P) covers are not unique, unlike the situation for projective covers. 相似文献
16.
If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of
S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far
there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S
I
(for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness.
Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats,
etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems.
S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494.
Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007). 相似文献
17.
António Veloso da Costa 《Semigroup Forum》2001,63(2):247-277
The main aim of this paper is to characterize the Green relations in the graph product of monoids. Necessary and sufficient
conditions for an element in a graph product of monoids to be idempotent, regular or completely regular, are established.
These characterizations immediately lead to decidability results. A new proof for the word problem is also presented.
May 22, 2000 相似文献
18.
Orthodox semidirect products and wreath products of monoids 总被引:17,自引:0,他引:17
Communicated by H.-J. Hoehnke 相似文献
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We explore connections between elementary equivalence of categories of acts over monoids and second-order equivalence of monoids.
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Translated from Algebrai Logika, Vol. 45, No. 6, pp. 687–709, November–December, 2006. 相似文献