The minimal closed monoids for the Galois connection End{\rm End}End-${\rm Con}$ (original) (raw)
Congruence lattices of connected monounary algebras
Algebra Universalis, 2020
The system of all congruences of an algebra (A, F) forms a lattice, denoted {{\\,\\mathrm{Con}\\,}}(A, F).Further,thesystemofallcongruencelatticesofallalgebraswiththebasesetAformsalattice. Further, the system of all congruence lattices of all algebras with the base set A forms a lattice.Further,thesystemofallcongruencelatticesofallalgebraswiththebasesetAformsalattice\\mathcal {E}_A.Wedealwithmeet−irreducibilityin. We deal with meet-irreducibility in.Wedealwithmeet−irreducibilityin\\mathcal {E}_AforagivenfinitesetA.Allmeet−irreducibleelementsoffor a given finite set A. All meet-irreducible elements offoragivenfinitesetA.Allmeet−irreducibleelementsof\\mathcal {E}_Aarecongruencelatticesofmonounaryalgebras.Sometypesofmeet−irreduciblecongruencelatticeswerealreadydescribed.Inthecasewhenamonounaryalgebra(A,f)isconnected,weprovenecessaryandsufficientconditionunderwhichare congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were already described. In the case when a monounary algebra (A, f) is connected, we prove necessary and sufficient condition under whicharecongruencelatticesofmonounaryalgebras.Sometypesofmeet−irreduciblecongruencelatticeswerealreadydescribed.Inthecasewhenamonounaryalgebra(A,f)isconnected,weprovenecessaryandsufficientconditionunderwhich{{\\,\\mathrm{Con}\\,}}(A, f)isisis\\wedge −irreducible.[GeneralizedquasiordersandtheGaloisconnection-irreducible.
[Generalized quasiorders and the Galois connection−irreducible.GeneralizedquasiordersandtheGaloisconnection{\textbf {End}}$$–$$\varvec{{{\,\textrm{gQuord}\,}}}$$
Algebra universalis, 2024
Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) have the property that an n-ary operation f preserves , i.e., f is a polymorphism of , if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves , i.e., it is an endomorphism of. We introduce a wider class of relations-called generalized quasiorders-of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End-gQuord, i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.
The lattice of congruence lattices of algebras on a finite set
Algebra universalis
The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice E. We describe the atoms and coatoms. Each meet-irreducible element of E being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice E; in particular, we prove that E is tolerance-simple whenever |A| ≥ 4. * supported by Slovak VEGA grant 1/0063/14 * * This research started as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project, supported by the European Union, co-financed by the European Social Fund 113/173/0-2.
The Galois correspondence between subvariety lattices and monoids of hipersubstitutions
Discussiones Mathematicae - General Algebra and Applications, 2000
Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigroups), and study the complete sublattices of this lattice obtained through the Galois correspondence.
Generalized Quasiorders and the Galois Connection End-gQuord
arXiv (Cornell University), 2023
Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) ϱ have the property that an n-ary operation f preserves ϱ, i.e., f is a polymorphism of ϱ, if and only if each translation (i.e., unary polynomial function obtained from f by substituting constants) preserves ϱ, i.e., it is an endomorphism of ϱ. We introduce a wider class of relations-called generalized quasiorders-of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End − gQuord, i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.
Endoprimal distributive lattices are endodualisable
Algebra Universalis, 1995
L. M/trki and R. P6schel have characterised the endoprimal distributive lattices as those which are not relatively complemented. The theory of natural dualities implies that any finite algebra A on which the endomorphisms of A yield a duality on the quasivariety DSP(A) is necessarily endoprimal. This note investigates endodualisability for finite distributive lattices, and shows, in a manner which elucidates Mfirki and P6schel's proof, that it is equivalent to endoprimality. Let (A; F) be an algebra and denote its endomorphism monoid by End A. Then A is said to be endoprimal if every finitary function on A commuting with each element of End A is a term function, and k-endoprimal (k >_ 1) if every function of arity not greater than k commuting with all endomorphisms is a term function. In [15], L. Mfirki and R. P6schel proved that the following statements are equivalent for a non-trivial distributive lattice (L; v, A): (1) L is endoprimal, and (2) L is not relatively complemented. The contrapositive of the implication (1) ~ (2) is easy (by exploiting relative complementation, as a ternary function). As Mfirki and P6schel point out, their result shows that an endoprimal algebra need not generate a quasivariety with the kind of nice structural properties that are obtained when primality is extended in other ways, notably to quasiprimality (see [16]). Thus the occurrence and role of endoprimal algebras seems a little mysterious, and examples are rather scarce. However there is one situation in which endoprimal algebras arise very naturally, through duality theory. When we refer to a duality, we shall always mean a natural duality in the sense of Davey and Werner. For our purposes, the recent survey [4] is the most convenient reference. A finite algebra A is said to be endodualisable if End A yields a duality on d = I~P(A), in the sense defined in [4], Section 2. If this
Congruence lattices of algebras of fixed similarity type. I
Pacific Journal of Mathematics, 1979
We prove that if V is any infinite-dimensional vector space over any uncountable field F, then the congruence lattice (=subspace lattice) of V cannot be represented as a congruence lattice (of any algebra) without using at least | F \ operations. This refutes a long-standing conjecture-that one binary operation would always suffice.
Lattice of quasiorders of monounary algebras
Miskolc Mathematical Notes, 2009
For an algebra A, the lattice Quord A of all quasiorders of A, i. e., of all reflexive and transitive relations compatible with all fundamental operations of A, is dealt with. In the present paper we prove that if A is a monounary algebra, then Quord A is distributive if and only if it is modular and we find necessary and sufficient conditions for A under which Quord A is distributive.
Mathematika, 1999
A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green's relations f and Jf coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids. PROPOSITION