Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property (SDC). Our main result is the description of finite lattices and semilattices with Property (SDC): we prove that a finite lattice has Property (SDC) if and only if it is a Boolean lattice, and a finite semilattice has Property (SDC) if and only if it is distributive.
Equations over linearly ordered semilattices are studied. For any equation $t(X)=s(X)$ we find irreducible components of its solution set and compute the average number of irreducible components of all equations in $n$ variables.
Equations over linearly ordered semilattices are studied. For any equation $t(X)=s(X)$ we find irreducible components of its solution set and compute the average number of irreducible components of all equations in $n$ variables.
We study centralizer clones of finite lattices and semilattices. For semilattices, we give two characterizations of the centralizer and also derive formulas for the number of operations of a given essential arity in the centralizer. We also characterize operations in the centralizer clone of a distributive lattice, and we prove that the essential arity of operations in the centralizer is bounded for every finite (possibly nondistributive) lattice. Using these results, we present a simple derivation for the centralizers of clones of Boolean functions.
As well-known, in a finitary algebraic structure the set $Γ$ of all the non-generators is the intersection of all the maximal proper substructures. In particular, $Γ$ is a substructure. We show that the corresponding statements hold for complete semilattices but fail for complete lattices, when as the notion of substructure we take complete subsemilattices and complete sublattices, respectively.
We construct a diagram D, indexed by a finite partially ordered set, of finite Boolean semilattices and (v,0,1)-embeddings, with top semilattice $2^4$, such that for any variety V of algebras, if D has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra $U$ in V such that the congruence lattice of $U$ contains, as a 0,1-sublattice, the five-element modular nondistributive lattice $M_3$. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive. Using earlier work of K.A. Kearnes an…
We attach to each $\langle 0, \vee \rangle$-semilattice a graph $\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph $\boldsymbol{G}_{\boldsymbol{S}}$ both when $\boldsymbol{S}$ is a join-semilattice and when it is a lattice. We call a $\langle 0, \vee \rangle$-semilattice $\boldsymbol{S}$ particle provided that the set of its join-irreducible elements join-generates $\boldsymbol{S}$ and it satisfies DCC. We prove that the congruence lattice of a parti…
Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an $n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp.
We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.
Part I proved that for every quasivariety K of structures (which may have both operations and relations) there is a semilattice S with operators such that he lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety Q such that the lattice of theories of Q is isomorphic to Con(S,+,0). We prove that if S is a semilattice having both 0 and 1 with a group G of operators acting on S, and each operator in G fixes both 0 and 1, then t…
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic problems.
L. P. Belluce, A. Di Nola and B. Gerla established a connection between MV-algebras and (dually) lattice ordered semirings by means of so-called coupled semirings. A similar connection was found for basic algebras and semilattice ordered right near semirings by the authors. The aim of this paper is to derive an analogous connection for orthomodular lattices and certain semilattice ordered near semirings via so-called coupled right orthosemirings.
For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.
We extend Wallman's classic duality from lattice bases to semilattice subbases and from compact to locally closed compact spaces. Moreover, we make this duality functorial via appropriate relational morphisms.
We study ideals and filters of posets and of pseudocomplemented posets and show a version of the Separation Theorem, known for ideals and filters in lattices and semilattices, within this general setting. We extend the concept of a *-ideal already introduced by Rao for pseudocomplemented distributive lattices and by Talukder, Chakraborty and Begum for pseudocomplemented semilattices to pseudocomplemented posets. We derive several important properties of such ideals. Especially, we explain connections between prime filters, ultrafilters, filters satisfying the *-condition and dense elements. Fi…
For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest cardinality of a semilattice in Conc(V1) which is not in Conc(V2) if it exists, infinity otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties V1 and V2, crit(V1,V2) is either finite, or aleph_n for some natural number n, or infinity. We also find two finitely generated modu…
A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, anti-isomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which …
For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex geometry, then Reg(P,f) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P,f) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For …
For P a poset or lattice, let Id(P) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect that Id(P) is always, and id(P) often, "essentially larger" than P. In the first vein, we find that a poset P admits no "<"-respecting map (and so in particular, no one-to-one isotone map) from Id(P) into P, and, going the other way, that an upper semilattice S admits no semilattice homomorphism from any subsemilattice of itself onto Id(S). The slightly smaller object id(P) is k…
We denote by Conc(A) the semilattice of all finitely generated congruences of an (universal) algebra A, and we define Conc(V) as the class of all isomorphic copies of all Conc(A), for A in V, for any variety V of algebras. Let V and W be locally finite varieties of algebras such that for each finite algebra A in V there are, up to isomorphism, only finitely many B in W such that A and B have isomorphic congruence lattices, and every such B is finite. If Conc(V) is not contained in Conc(W), then there exists a semilattice of cardinality aleph 2 in Conc(V)-Conc(W). Our result extends to quasivar…
Various authors have investigated properties of the star order (introduced by M.P. Drazin in 1978) on algebras of matrices and of bounded linear operators on a Hilbert space. Rickart involution rings (*-rings) are a certain algebraic analogue of von Neumann algebras, which cover these particular algebras. In 1983, M.F. Janowitz proved, in particular, that, in a star-ordered Rickart *-ring, every pair of elements bounded from above has a meet and also a join. However, the latter conclusion seems to be based on some wrong assumption. We show that the conclusion is nevertheless correct, and provi…
