Morita duality for monoids

In this paper Morita duality for monoids is introduced. Necessary and sufficient conditions for two monoids S and T to be Morita dual are given. Moreover, it is shown that if S and T are Morita dual monoids, then S and U are Morita dual if and only if T and U are Morita equivalent. In addition, every finite monoid having Morita duality is selfdual and even reflexive.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

可分解半群的Morita等价

设S,R是可分解半群.记US-FAct={sM∈S-Act|SM=M且SHoms(S,M)≌M],给出了范畴US-FAct与UR-FAct等价的刻划;S分别强Morita等价于一个夹层半群、局部单位半群、幺半群和群的条件;S是完全单半群当且仅当S强Morita等价于一个群且对任何指标集I,S SHoms(S,i∈I S)→i∈I S,s t·f→(st)f,是同构.     

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R., _S P _R,R Q _S ,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ _S = {(s ₁, s ₂) ∈S×S|ss ₁ = ss ₂, ∀s∈S}, S' = S/ζ _S and US-FAct = { _S M∈S− Act |SM = M and SHom _S (S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom _S (S, ∐ _i∈I S) →∐ _i∈I S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism. The research is partially supported by a UGC(HK) grant #2160092. Project is supported by the National Natural Science Foundation of China     

Semigroups embeddable in hyperplane face monoids

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.     

Characterizations of Morita equivalent inverse semigroups

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C^∗-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.     

On the torsion theories of Morita equivalent rings

We generalize the well-known fact that for a pair of Morita equivalent ringsR andS their maximal rings of quotients are again Morita equivalent: If _n (M) denotes the torsion theory cogenerated by the direct sum of the firstn+1 injective modules forming part of the minimal injective resolution ofM then _n (R)= _n (S) where is the category equivalenceR-ModS-Mod. Consequently the localized ringsR _{n (R)} andS n _(S) are Morita equivalent.     

The structure of 0-bisimple strongly E *-unitary inverse monoids

We introduce a class of strongly E ^*-unitary inverse semigroups S _i (G, P) (i = 1,2) determined by a group G and a submonoid P of G and give an embedding theorem for S _i (G, P). Moreover we characterize 0-bisimple strongly E ^*-unitary inverse monoids and 0-bisimple strongly F ^*-inverse monoids by using S _i (G, P).     

On monoids over which all strongly flat cyclic right acts are projective

A new characterization of monoids over which all strongly flat cyclic right acts are projective (projective generators, free) is given. This research has been supported by the Estonian Science Foundation, Grant No. 930.     

Endomorphism monoids of acts over monoids

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.     

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest  N∈N₀∪{∞} having the following property: if  n∈N and  a₁,…,a_n∈H are such that b divides  a₁⋅…⋅a_n, then b already divides a subproduct of a₁⋅…⋅a_n consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.