Sheffer operation in relational systems

Relational Systems ◽

AbstractThe concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative, then the induced binary relation is antisymmetric. Moreover, commutative Sheffer groupoids form a congruence distributive variety. We characterize symmetry, antisymmetry and transitivity of binary relations by identities and quasi-identities satisfied by an assigned Sheffer operation. The concepts of twist products of relational systems and of Kleene relational systems are introduced. We prove that every directed relational system can be embedded into a directed relational system with involution via the twist product construction. If the relation in question is even transitive, then the directed relational system can be embedded into a Kleene relational system. Any Sheffer operation assigned to a directed relational system $${\mathbf {A}}$$ A with involution induces a Sheffer operation assigned to the twist product of $${\mathbf {A}}$$ A .

SOLUTION TO A PROBLEM OF KUBLANOVSKY AND SAPIR

International Journal of Algebra and Computation ◽

10.1142/s0218196701000644 ◽

2001 ◽

Vol 11 (04) ◽

pp. 489-495 ◽

Cited By ~ 1

Author(s):

DEJAN DELIĆ

Keyword(s):

Variety Of Algebras ◽

Finitely Based ◽

Distributive Variety ◽

Finite Language ◽

Simple Algebras ◽

We construct a finitely based congruence-distributive variety of algebras in a finite language whose set of subalgebras of finite simple algebras is non-recursive.

CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196712500531 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250053 ◽

Cited By ~ 1

Author(s):

PIERRE GILLIBERT ◽

MIROSLAV PLOŠČICA

Keyword(s):

Necessary Condition ◽

Finitely Generated ◽

Distributive Variety ◽

Finite Lattices ◽

Congruence Lattices ◽

Congruence Distributive ◽

Special Congruence

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.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Locally finite affine complete varieties

10.1017/s1446788700000720 ◽

1997 ◽

Vol 62 (2) ◽

pp. 141-159 ◽

Cited By ~ 3

Author(s):

Kalle Kaarli

Keyword(s):

Distributive Variety ◽

Locally Finite ◽

Complete Variety ◽

Finite Affine ◽

Near Unanimity Term ◽

Affine Complete ◽

Congruence Distributive ◽

Near Unanimity

AbstractThe main results of the paper are the following: 1. Every locally finite affine complete variety admits a near unanimity term; 2. A locally finite congruence distributive variety is affine complete if and only if all its algebras with no proper subalgebras are affine complete and the variety is generated by one of such algebras. The first of these results sharpens a result of McKenzie asserting that all locally finite affine complete varieties are congruence distributive. The second one generalizes the result by Kaarli and Pixley that characterizes arithmetical affine complete varieties.

A Subalgebra Intersection Property for Congruence Distributive Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-023-2 ◽

2009 ◽

Vol 61 (2) ◽

pp. 451-464 ◽

Cited By ~ 10

Author(s):

Matthew A. Valeriote

Keyword(s):

Constraint Satisfaction ◽

Constraint Satisfaction Problem ◽

Finite Algebra ◽

Intersection Property ◽

Distributive Variety ◽

Abstract. We prove that if a finite algebra Agenerates a congruence distributive variety, then the subalgebras of the powers of Asatisfy a certain kind of intersection property that fails for finite idempotent algebras that locally exhibit affine or unary behaviour. We demonstrate a connection between this property and the constraint satisfaction problem.

Absolute Retracts and Amalgamation in Certain Congruence Distributive Varieties

Canadian Mathematical Bulletin ◽

10.4153/cmb-1989-044-x ◽

1989 ◽

Vol 32 (3) ◽

pp. 309-313 ◽

Cited By ~ 5

Author(s):

Peter Jipsen ◽

Henry Rose

Keyword(s):

Direct Products ◽

Distributive Variety ◽

Absolute Retracts ◽

Amalgamation Class ◽

AbstractIt is shown that if V is a congruence distributive variety whose members have one element subalgebras, then the class of absolute retracts of V is closed under direct products. If V is residually small, then a characterisation of the amalgamation class of V is given.

Finitely Related Algebras In Congruence Distributive Varieties Have Near Unanimity Terms

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-087-3 ◽

2013 ◽

Vol 65 (1) ◽

pp. 3-21 ◽

Cited By ~ 18

Author(s):

Libor Barto

Keyword(s):

Constraint Satisfaction Problem ◽

Distributive Variety ◽

Term Operation ◽

Relational Structures ◽

Near Unanimity Term ◽

Finite Set ◽

Congruence Distributive ◽

Near Unanimity ◽

Finitely Related

Abstractwe show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. as a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. this consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

The syntax and semantics of entailment in duality theory

Journal of Symbolic Logic ◽

10.2307/2275875 ◽

1995 ◽

Vol 60 (4) ◽

pp. 1087-1114 ◽

Cited By ~ 15

Author(s):

B. A. Davey ◽

M. Haviar ◽

H. A. Priestley

Keyword(s):

Duality Theory ◽

Finite Algebra ◽

Partial Functions ◽

Distributive Variety ◽

Positive Formula ◽

Primitive Positive Formula ◽

Finite Set ◽

Clone Theory ◽

AbstractBoth syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

Iterative separation in distributive congruence lattices

Mathematica Slovaca ◽

10.2478/s12175-009-0119-2 ◽

2009 ◽

Vol 59 (2) ◽

Cited By ~ 1

Author(s):

Miroslav Ploščica

Keyword(s):

Close Connection ◽

Subdirectly Irreducible ◽

Finitely Generated ◽

Distributive Variety ◽

Algebraic Lattices ◽

Congruence Lattices ◽

Congruence Distributive ◽

Subdirectly Irreducible Algebras

AbstractIn [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties.

The existence of a near-unanimity term in a finite algebra is decidable

Journal of Symbolic Logic ◽

10.2178/jsl/1245158096 ◽

2009 ◽

Vol 74 (3) ◽

pp. 1001-1014 ◽

Cited By ~ 7

Author(s):

Miklós Maróti

Keyword(s):

Finite Algebra ◽

Natural Duality ◽

Distributive Variety ◽

Term Operation ◽

Near Unanimity Term ◽

Congruence Distributive ◽

Near Unanimity

AbstractWe prove that it is decidable of a finite algebra whether it has a near-unanimity term operation, which settles a ten-year-old problem. As a consequence, it is decidable of a finite algebra in a congruence distributive variety whether it admits a natural duality.

PRINCIPAL AND SYNTACTIC CONGRUENCES IN CONGRUENCE-DISTRIBUTIVE AND CONGRUENCE-PERMUTABLE VARIETIES

Journal of the Australian Mathematical Society ◽

10.1017/s144678870800061x ◽

2008 ◽

Vol 85 (1) ◽

pp. 59-74 ◽

Cited By ~ 7

Author(s):

BRIAN A. DAVEY ◽

MARCEL JACKSON ◽

MIKLÓS MARÓTI ◽

RALPH N. MCKENZIE

Keyword(s):

Additional Assumption ◽

Finitely Generated ◽

Distributive Variety ◽

Congruence Permutable Varieties ◽

AbstractWe give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite.