Algebraic Lattices Research Articles

Isometrical Embeddings of Lattices into Geometric Lattices

A lattice is said to be finite height generated if it is complete and every element is the join of some elements of finite height. Extending former results by Gratzer and Kiss (Order 2:351–365, 1986) on finite lattices, we prove that every finite height generated algebraic lattice that has a pseudorank function is isometrically embeddable into a geometric lattice.

A note on representation of lattices by weak congruences

A weak congruence is a symmetric, transitive, and compatible relation. An element u of an algebraic lattice L is Δ-suitable if there is an isomorphism κ from L to the lattice of weak congruences of an algebra such that κ(u) is the diagonal relation. Some conditions implying the Δ-suitability of u are presented.

CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.

Semilattices of definable subalgebras. II

In studying derived objects on universal algebras, such as automorphisms, endomorphisms, congruences, subalgebras, etc., we are naturally interested in those that can be defined by the means of the universal algebras themselves (i.e., are definable in one sense or another)—in particular, in which part of all relevant derived objects is constituted by these. It is proved that for any algebraic lattice L and any of its 0-1-lower subsemilattices L 0 ⊆ L 1 ⊆ L 2, there exist a universal algebra $ \mathcal{A} $ and an isomorphism φ of the lattice L onto the lattice Sub $ \mathcal{A} $ such that φ(L 0) = OFSub $ \mathcal{A} $ , φ(L 1) = POFSub $ \mathcal{A} $ , φ(L 2) = FSub $ \mathcal{A} $ , and PFSub $ \mathcal{A} $ = FSub $ \mathcal{A} $ .

On finite retract lattices of monounary algebras

Abstract For a monounary algebra (A, f) we denote R ∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R ∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R ∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras.

Some properties of retract lattices of monounary algebras

Abstract Necessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ∅(A, f) is complemented. In this case R ∅(A, f) forms a boolean lattice.

THE LATTICE OF ALTER EGOS

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.

On delta-suitable elements in algebraic lattices

In this paper we introduce new methods and results that can lead to the solution of a long-standing open problem of the representation of algebraic lattices by weak congruences. Here known criteria for ?-suitable elements in algebraic lattices are generalized.

Filters in (Quasiordered) Semigroups and Lattices of Filters

A filter in a semigroup is a subsemigroup whose complement is an ideal. (Alternatively, in a quasiordered semigroup, a slightly more general definition can be given.) We prove a number of results related to filters in a semigroup and the lattice of filters of a semigroup. For instance, we prove that every complete algebraic lattice can be the lattice of filters of a semigroup. We prove that every finite semigroup is a homomorphic image of a finite semigroup whose lattice of filters is boolean and which belongs to the pseudovariety generated by the original semigroup. We describe filter lattices of some well-known semigroups such as full transformation semigroups of finite sets (which are three-element chains) and free semigroups (which are boolean).

Representation of Nelson algebras by rough sets determined by quasiorders

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if $R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.

Pre-ideals of Basic Algebras

Basic algebras are a generalization of MV-algebras, also including orthomodular lattices and lattice effect algebras. A pre-ideal of a basic algebra is a non-empty subset that is closed under the addition ⊕ and downwards closed with respect to the underlying order. In this paper, we study the pre-ideal lattices of algebras in a particular subclass of basic algebras which are closer to MV-algebras than basic algebras in general. We also prove that finite members of this subclass are exactly finite MV-algebras.

Complemented quasiorder lattices of monounary algebras

We characterize monounary algebras whose lattices of quasiorders are complemented. As a consequence, monounary algebras with Boolean quasiorder lattices are described.

Algebraic lattices and locally finitely presentable categories

We show that subobjects and quotients respectively of any object K in a locally finitely presentable category form an algebraic lattice. The same holds for the internal equivalence relations on K. In fact, these results turn out to be—at least in the case of subobjects—nothing but simple consequences of well known closure properties of the classes of locally finitely presentable categories and accessible categories, respectively. We thus get a completely categorical explanation of the well known fact that the subobject- and congruence lattices of algebras in finitary varieties are algebraic. Moreover we also obtain new natural examples: in particular, for any (not necessarily finitary) polynomial set-functor F, the subcoalgebras of an F-coalgebra form an algebraic lattice; the same holds for the lattices of regular congruences and quotients of these F-coalgebras.

On the prevariety of perfect lattices

We call a complete lattice perfect if it is a sublattice of a lattice of the form Sp(A), where A is an algebraic lattice and Sp(A) stands for the lattice of algebraic subsets of A.

Special Lattice of Rough Algebras

This paper deals with the study of the special lattices of rough algebras. We discussed the basic properties such as the rough distributive lattice; the rough modular lattice and the rough semi-modular lattice etc., some results of lattice are generalized in this paper. The modular lattice of rough algebraic structure can provide academic base and proofs to analyze the coverage question and the reduction question in information system.

IRREDUCIBLE ELEMENTS IN ALGEBRAIC LATTICES

α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements, for an algebraic lattice to satisfy the complete distributivity.

Graphical algebras — a new approach to congruence lattices

In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Gratzer and Schmidt’s result that any algebraic lattice occurs as a congruence lattice.

Rings and Gödel algebras

In this paper, we study those rings whose semiring of ideals can be given the structure of a Godel algebra. Such rings are called Godel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Godel ring R is a subdirect product of prime Godel rings R i , and the Godel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into $${{\prod_{i}}Id(R_{i})}$$ , where each Id(R i ) is the algebraic lattice of ideals of R i that can be equipped with the structure of a Godel algebra. We see that the mapping associating to each Godel ring its Godel algebra of ideals is functorial from the category of Godel rings with epimorphisms into the full subcategory of frames whose objects are Godel algebras and whose morphisms are complete epimorphisms.

Using coloured ordered sets to study finite-level full dualities

We consider all the full dualities for the class of finite bounded distributive lattices that are based on the three-element chain 3. Under a natural quasi-order, these full dualities form a doubly algebraic lattice \({\mathcal{F}_{\underline{3}}}\). Using Priestley duality, we establish a correspondence between the elements of \({\mathcal{F}_{\underline{3}}}\) and special enriched ordered sets, which we call ‘coloured ordered sets’. We can then use combinatorial arguments to show that the lattice \({\mathcal{F}_{\underline{3}}}\) has cardinality \({2^{\aleph_{0}}}\) and is non-modular. This is the first investigation into the structure of an infinite lattice of finite-level full dualities.

More Sublattices of the Lattice of Local Clones

We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that: For example, the lattice \(M_{2^\omega}\) is a sublattice of the local clone lattice.