BCC-algebras and residuated partially-ordered groupoid

The aim of the paper is to investigate the relationship between BCC-algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC-algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

The subvariety of commutative residuated lattices represented by twist-products

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety ${\mathbb{K}}$ of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of ${\mathbb{K}}$ , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in ${\mathbb{K}}$ , and we analyze the subvariety of representable algebras in ${\mathbb{K}}$ . Finally, we consider some specific class of bounded integral commutative residuated lattices ${\mathbb{G}}$ , and for each fixed element ${{\bf L} \in \mathbb{G}}$ , we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras.     

On involutive FL e -monoids

The paper deals with involutive FL _e -monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FL _e -monoids over lattices are exactly involutive FL _e -algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FL _e -monoids, along with a new construction method, called twin-rotation. Some classes of finite involutive FL _e -chains are classified by using the notion of rank of involutive FL _e -chains, and a kind of duality is developed between positive and non-positive rank algebras. As a side effect, it is shown that the substructural logic IUL plus t ? f does not have the finite model property.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

Normalizers,Centralizers and Action Representability in Semi-Abelian Categories

We introduce the notion of normalizer as motivated by the classical notion in the category of groups. We show for a semi-abelian category ? that the following conditions are equivalent:

1. ? is action representable and normalizers exist in ?;
2. the category Mono(?) of monomorphisms in ? is action representable;
3. the category ?² of morphisms in ? is action representable;
4. for each category \(\mathbb {D}\) with a finite number of morphisms the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable.

Moreover, when in addition ? is locally well-presentable, we show that these conditions are further equivalent to:

1. ? satisfies the amalgamation property for protosplit normal monomorphism and ? satisfies the axiom of normality of unions;
2. for each small category \(\mathbb {D}\) , the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable.

We also show that if ? is homological, action accessible, and normalizers exist in ?, then ? is fiberwise algebraically cartesian closed.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

On finite elements in f-algebras and in product algebras

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell $ -algebras, preferably in $f$ -algebras, and in product algebras. The additional structure of an associative multiplication leads to new questions and some new properties concerning the collections of finite, totally finite and self-majorizing elements. In some cases the order ideal of finite elements is a ring ideal as well. It turns out that a product of elements in an $f$ -algebra is a finite element if at least one factor is finite. If the multiplicative unit exists, the latter plays an important role in the investigation of finite elements. For the product of special $f$ -algebras an element is finite in the algebra if and only if its power is finite in the product algebra.     

On a class of σ-space with special nets

Strong CP(HCP)-netted spaces are defined and some properties are shown. In particular, the following results are shown.

1. A submetrizable space is strong CP(HCP)-netted provided that the space admits a perfect map onto a strong CP(HCP)-netted space.
2. The image of a strong CP(HCP)-netted space under a perfect map is strong CP(HCP)-netted space.
3. A stratifiable space is strong HCP-netted if the space has a countable closed cover consisting of strong HCP-netted subspaces.

     

Invariants of Lie superalgebras acting on associative algebras

LetR be a semiprime algebra over a fieldK acted on by a finite-dimensional Lie superalgebraL. The purpose of this paper is to prove a series of going-up results showing how the structure of the subalgebra of invariantsR ^Lis related to that ofR. Combining several of our main results we have: Theorem: Let R be a semiprime K-algebra acted on by a finite-dimensional nilpotent Lie superalgebra L such that if characteristic K=p then L is restricted and if characteristic, K=0 then L acts on R as algebraic derivations and algebraic superderivations.

1. If R^L is right Noetherian, then R is a Noetherian right R^L-module. In particular, R is right Noetherian and is a finitely generated right R^L-module.
2. If R^L is right Artinian, then R is an Artinian right R^L-module. In particular, R is right Artinian and is a finitely generated right R^L-module.
3. If R^L is finite-dimensional over K then R is also finite-dimensional over K.
4. If R^L has finite Goldie dimension as a right R^L-module, then R has finite Goldie dimension as a right R-module.
5. If R^L has Krull dimension α as a right R^L-module, then R has Krull dimension α as a right R^L-module. Thus R has Krull dimension at most α as a right R-module.
6. If R is prime and R^L is central, then R satisfies a polynomial identity.
7. If L is a Lie algebra and R^L is central, then R satisfies a polynomial identity.

We also provide counterexamples to many questions which arise in view of the results in this paper.