arXiv subjects
Publications and source records attributed to Paolo Lipparini.
We study contact posets and show that every contact poset can be embedded into a Boolean poset with overlap contact relation. Contact posets and (nonadditive) contact semilattices have the superamalgamation property, Fra\"ıssé limits and model completion. Some results apply to event structures with binary conflict, as introduced in computer science.
The superamalgamation property is a strong form of the amalgamation property which applies to ordered structures; it has found many applications in algebraic logic. We show that superamalgamation has some interest also from the pure model-theoretical point of view. Under a completion assumption, we prove that the superamalgamation property for some class of ordered structures implies strong amalgamation for classes with added operations, including isotone, idempotent, extensive, antitone and closure operations. Thus, for example, partially ordered sets, semilattices, lattices, Boolean algebras and Heyting algebras with an isotone extensive operation (or an operation as above) have the strong amalgamation property. The theory of join semilattices with a closure operation has model completion. The set of universal consequences of the theory of Boolean algebras (or posets, semilattices, distributive lattices) with a closure or isotone, etc., operation is decidable.
We consider the situation in which some class of structures has the Strong Amalgamation Property (SAP) with the further requirement that the amalgamating structure can be taken over the set theoretical union of (the images of) the domains of the structures to be amalgamated. We call this property SAPU. The main advantage of SAPU over SAP is that there are many preservation theorems showing that we can merge different theories with SAPU still obtaining a theory with SAPU, hence with SAP. In particular, we get SAPU for various theories with many binary relations, each relation satisfying any set of properties chosen among transitivity, reflexivity, symmetry, antireflexivity, antisymmetry. We may also add unary operations, possibly satisfying some coarseness, isotonicity and closure conditions. SAPU is not limited to relational theories: the varieties defining the most usual Maltsev conditions in universal algebra have SAPU. Other examples include bounded directoids, order algebras and various generalization.
The theory of two binary relations has the strong amalgamation property when the first relation is assumed to be coarser than the second relation, and each relation satisfies a chosen set of properties from the following list: transitivity, reflexivity, symmetry, antireflexivity and antisymmetry. The amalgamation property is maintained when we add families of unary operations preserving all the relations. As a consequence, we get the existence of Fra\"ıssé limits for classes of finite structures. The results fail, for general comparability conditions, when three or more binary relations are taken into account, or when we add an operation preserving just one relation.
We say that an idempotent term $t$ is an exact-$m$-majority term if $t$ evaluates to $a$, whenever the element $a$ occurs exactly $m$ times in the arguments of $t$, and all the other arguments are equal. If $m<n$ and some variety $\mathcal V$ has an $n$-ary exact-$m$-majority term, then $\mathcal V$ is congruence modular. For certain values of $n$ and $m$, for example, $n=5$ and $m=3$, the existence of an $n$-ary exact-$m$-majority term neither implies congruence distributivity, nor congruence permutability.
If $X$ is a closure space with closure $K$, we consider the semilattice $(\mathcal P(X), \cup)$ endowed with further relations $ x \sqsubseteq y_1, y_2, \dots, y_n$ (a distinct $n+1$-ary relation for each $n \geq 1$), whose interpretation is $x \subseteq Ky_1 \cup Ky_2 \cup \dots \cup Ky_n $. We present axioms for such "multi-argument specialization semilattices" and show that this list of axioms is complete for substructures, namely, every model satisfying the axioms can be embedded into some structure originated by some closure space as in the previous sentence. We also provide a canonical embedding of a multi-argument specialization semilattice into (the reduct of) some closure semilattice.
We solve some problems about relative lengths of Maltsev conditions, in particular, we give an affirmative answer to a classical problem raised by A. Day more than fifty years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms $t_0, \dots, t_n$ witnessing congruence distributivity it is possible to construct terms $u_0, \dots, u _{2n-1} $ witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when $n$ is even. We also deal with other kinds of terms, such as alvin, Gumm, Pixley, directed, specular, mixed and defective. All the results hold when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.
A specialization semilattice is a structure which can be embedded into $(\mathcal P(X), \cup, \sqsubseteq )$, where $X$ is a topological space, $ x \sqsubseteq y$ means $x \subseteq Ky$, for $x,y \subseteq X$, and $K$ is closure in $X$. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In general, closure is not expressible in a specialization semilattice. On the other hand, we show that every specialization semilattice can be canonically embedded into a "principal" specialization semilattice in which closure can be actually defined.
A specialization semilattice is a join semilattice together with a coarser preorder $ \sqsubseteq $ satisfying an appropriate compatibility condition. If $X$ is a topological space, then $(\mathcal P(X), \cup, \sqsubseteq )$ is a specialization semilattice, where $ x \sqsubseteq y$ if $x \subseteq Ky$, for $x,y \subseteq X$, and $K$ is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. For short, the notion is useful since it allows us to consider a relation of "being generated by" with no need to require the existence of an actual "closure" or "hull", which might be problematic in certain contexts. In a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. Here we describe the universal embedding of a specialization semilattice into an additive closure semilattice. We notice that a categorical argument guarantees the existence of universal embeddings in many parallel situations.
We devise a condition strictly between the existence of an $n$-ary and an $n{+}1$-ary near-unanimity term. We evaluate exactly the distributivity and modularity levels implied by such a condition.
We introduce a general notion of covering property, of which many classical definitions are particular instances. Notions of closure under various sorts of convergence, or, more generally, under taking kinds of accumulation points, are shown to be equivalent to a covering property in the sense considered here (Theorem 3.9). Conversely, every covering property is equivalent to the existence of appropriate kinds of accumulation points for arbitrary sequences on some fixed index set (Corollary 3.5). We discuss corresponding notions related to sequential compactness, and to pseudocompactness, or, more generally, properties connected with the existence of limit points of sequences of subsets. In spite of the great generality of our treatment, many results here appear to be new even in very special cases, such as $D$-compactness and $D$-pseudocompactness, for $D$ an ultrafilter, and weak (quasi) $M$-(pseudo)-compactness, for $M$ a set of ultrafilters, as well as for $[β, α]$-compactness, with $β$ and $α$ ordinals.
We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. Let $X$ be a product of topological spaces. We prove that $X$ is sequentially compact if and only if all subproducts by $\leq \mathfrak s$ factors are sequentially compact. If $\mathfrak s = \mathfrak h$, then $X$ is sequentially compact if and only if all factors are sequentially compact and all but at most $<\mathfrak s$ factors are ultraconnected. We give a topological proof of the inequality ${\rm cf} \mathfrak s \geq \mathfrak h$. Recall that $\mathfrak s$ denotes the splitting number and $\mathfrak h$ the distributivity number. The product $X$ is Lindelöf if and only if all subproducts by $\leq ω_1 $ factors are Lindelöf. Parallel results are obtained for final $ ω_n$-compactness, $[ λ, μ]$-compactness, the Menger and Rothberger properties.
We provide conditions under which the union of two first-order theories has the amalgamation property.
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.
A folklore argument shows that Frattini's characterization of non-generators works in the framework of algebraic partially ordered sets. We provide characterizations of non-generators in arbitrary partially ordered sets. The validity of some characterizations is equivalent to Zorn's Lemma, hence to the Axiom of choice. We notice that working on closure spaces or posets provides no essential improvement.
The class of linearly ordered sets with one order preserving unary operation has the Strong Amalgamation Property (SAP). The class of linearly ordered sets with one strict order preserving unary operation has AP but not SAP. The class of linearly ordered sets with two order preserving unary operations has not AP. For every set $F$, the class of linearly ordered sets with an $F$-indexed family of automorphisms has SAP. Corresponding results are proved in the case of order reversing operations. Various subclasses of the above classes are considered and some model-theoretical consequences are presented.
A variety is a class of algebraic structures axiomatized by a set of equations. An equation is linear if there is at most one occurrence of an operation symbol on each side. We show that a variety axiomatized by linear equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds. Examples include most varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operations appear in the equations.
A. Mitschke showed that a variety with an $m$-ary near-unanimity term has Jónsson terms $t_0, \dots, t _{2m-4} $ witnessing congruence distributivity. We show that Mitschke's result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an $m$-ary near-unanimity term.