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and/or . We shall mention that several general theorems from literature can be applied to obtain results concerning [MATH] . For example, Theorem below (at least for the case when [MATH] is finite) may follow as a consequence of 30 , Theorem 9.4] 10 , Theorem 4.2] and 21 , Theorem 5.3.5] |
In the present paper, we give direct proofs of these facts (for finite and infinite [MATH] ’s). Our technique also leads to some new important results, see Theorem and Theorem |
Theorem 1 The variety [MATH] enjoys each of the following: (1) Finite schema axiomatizability (finite axiomatizability if [MATH] is finite). |
(2) Finite base property, i.e.,, (3) Decidable equational theory. (4) [MATH] is infinite. (5) Super amalgamation property. Theorem 2 |
Let [MATH] be any set, we have the following: If [MATH] then [MATH] is a two-element algebra, hence it is atomic. If [MATH] then the free algebra [MATH] is atomless. |
Theorem 3 Let [MATH] be any set. The only zero-dimensional elements in the free algebra [MATH] are the zero and the unit. To prove the above theorems, we use the normal forms defined in |
, these are generalizations of the normal forms introduced by J. Hintikka . We show that, for each satisfiable normal form in [MATH] , there is an algebra in [MATH] , whose base is finite, that witnesses the satisfiability of this form. |
3. Axioms and normal forms Here, we give an equational characterization for the class [MATH] . We get this characterization by deleting the cylindrifiers-commutativity axiom from the axioms defining the diagonal free cylindric algebras defined by A. Tarski , Definition 1.1.2] |
Definition 3.1 Let [MATH] be the class of all relativized diagonal free algebras of dimension [MATH] , i.e., the class consists of all algebras [MATH] , that satisfy the following equations for every [MATH] |
(Ax 0) The set of equations characterizing Boolean algebras for [MATH] (Ax 1) The set of equations defining [MATH] as an additive, closure and complemented operator: |
(Ax 1a) [MATH] (Ax 1b) [MATH] (Ax 1c) [MATH] Note that [MATH] . It is easy to check that each [MATH] satisfies the above axioms. Later, we will prove that the above is actually a characterization of the class [MATH] , i.e., [MATH] . Let [MATH] be any set of variables, then [MATH] is defined to be the set of all terms i... |
Now, we define normal forms in the signature of [MATH] . Then, we will show that each term in this signature can be rewritten equivalently as a Boolean joint of these normal forms. Let [MATH] and [MATH] be the grouped versions of [MATH] and [MATH] respectively. Empty product and empty sum are defined to be [MATH] and [... |
[EQUATION] where, for every [MATH] [MATH] if [MATH] and [MATH] otherwise. Definition 3.2 Let [MATH] be a finite set and let [MATH] be finite ordinal such that [MATH] . Let [MATH] , we define the following inductively. |
Normal forms of degree [MATH] [MATH] Normal forms of degree [MATH] [MATH] All normal forms, [MATH] The prove of the following theorem can be found in 44 , Lemma 4.9 and Theorem 4.10] |
Theorem 3.3 Let [MATH] and [MATH] be finite ordinals such that [MATH] . Let [MATH] be a finite set of variables. Then the following are true: |
(i) [MATH] (ii) For every [MATH] , if [MATH] then [MATH] (iii) Let [MATH] be such that [MATH] . Then there is a finite ordinal [MATH] and a non-empty finite set [MATH] of normal forms of degree [MATH] such that [MATH] |
Note that (i) and (ii) of the above theorem state that [MATH] forms a partition of the unit. The following definition introduces some notations that will be used in the proceeding sections. |
Definition 3.4 Let [MATH] and [MATH] be finite ordinals such that [MATH] . Let [MATH] be a finite set of variables. Let [MATH] and [MATH] . For each [MATH] , define |
[MATH] , and [MATH] 4. Finite schema axiomatizability, decidability, etc In this section, we fix finite ordinals [MATH] and [MATH] such that [MATH] , and we fix finite set [MATH] . Consider any normal form [MATH] . We will construct a unit [MATH] , on a finite base, such that |
[EQUATION] We do this inductively by constructing a finite sequence [MATH] (of length [MATH] ), and then we let [MATH] be the desired unit. Throughout the construction, whenever we add an element [MATH] , we label it by some normal form [MATH] . While constructing [MATH] , our target is to guarantee that each element s... |
To start, let [MATH] be an infinite set and let [MATH] be any entity. For any elements [MATH] , by writing [MATH] we mean the sequence, of length [MATH] , defined as follows: For each [MATH] [MATH] . For each [MATH] [MATH] . The sequence [MATH] is called the tail of the desired unit [MATH] |
Constructing [MATH] Let [MATH] be such that [MATH] . Let [MATH] . Define the label of the unique element in [MATH] as follows: [MATH] |
Suppose that, for some [MATH] , we are given the finite sets [MATH] [MATH] . Also, assume that we are given the labels of the elements in [MATH] |
Constructing [MATH] For every [MATH] , every [MATH] and every [MATH] , create an injective function [EQUATION] such that the ranges [MATH] of all of those functions are pairwise disjoint and [MATH] is still infinite, where [MATH] . Now, for every [MATH] , let |
[EQUATION] Let [MATH] . We extend the labels as follows: Let [MATH] [MATH] [MATH] and [MATH] . Suppose that [MATH] , define [MATH] |
The desired algebra: Finally, let [MATH] and [MATH] . Remember [MATH] . We call [MATH] the actual base of [MATH] Note that, for each [MATH] and each [MATH] , we have |
[EQUATION] Define the evaluation, [MATH] , of free variables into [MATH] as follows: For each [MATH] , let [MATH] . For every [MATH] and every term [MATH] , we write [MATH] if and only if [MATH] is in the interpretation of the term [MATH] in the full algebra [MATH] , under the evaluation [MATH] . Now, we prove that [MA... |
Lemma 4.1 [MATH] if and only if [MATH] Proof. Note that [MATH] , so the direction ( [MATH] ) is trivial. Now, we prove the non-trivial direction ( [MATH] ). Suppose that [MATH] . Then by (Ax 1a), it follows that [MATH] for each element [MATH] . For each [MATH] and each [MATH] , we define [MATH] as follows: |
Suppose that [MATH] is the smallest number for which [MATH] . Remember the facts [MATH] and [MATH] (by ( )). If [MATH] then we define [MATH] . Suppose [MATH] , then by Theorem 3.3 there is a unique normal form [MATH] such that [MATH] , so in this case we define [MATH] |
To finish, it is enough to prove the following. For each [MATH] and each [MATH] [EQUATION] We use induction on [MATH] . The choice of the evaluation guarantees that ( ) is true for every [MATH] when [MATH] . Suppose that ( ) holds for every [MATH] , for some [MATH] . Let [MATH] , we need to show that [MATH] . Let [MATH... |
(I) The first case is when [MATH] . In this case, [MATH] . By the choice of the evaluation [MATH] , it is easy to check that [EQUATION] |
Let [MATH] and let [MATH] . We need to prove the following. [EQUATION] Suppose that [MATH] . By Theorem 3.3 (iii), there is a finite set [MATH] such that [MATH] . We claim that there is [MATH] such that [MATH] . Suppose towards a contradiction that [MATH] for every [MATH] . Then, by the additivity of the operator [MATH... |
Conversely, let [MATH] be such that [MATH] and [MATH] . Suppose that [MATH] , then by induction hypothesis and Theorem 3.3 (ii) we have [MATH] . Again, by Theorem 3.3 [MATH] . Thus, by axiom (Ax 1b), [MATH] as desired. Suppose that [MATH] and [MATH] are different. By the construction, there exists [MATH] such that [MAT... |
Thus, we have shown that ( ) is true for every [MATH] and every [MATH] . Therefore, by ( ) and ( ), we have [MATH] , as desired. |
(II) Now, suppose that [MATH] . Again the choice of the evaluation guarantees the following. [EQUATION] Suppose that [MATH] for some [MATH] . Let [MATH] be such that [MATH] . Then, by a similar argument to the one used in the above item, one can see that |
[EQUATION] By the construction of [MATH] and since [MATH] , there exists an element [MATH] such that [MATH] is the smallest number for which [MATH] [MATH] and [MATH] . Thus, by axioms (Ax 1a) and (Ax 1c), and Theorem 3.3 , it follows that [MATH] . Since [MATH] , then [MATH] . Now, by ( ) it follows that |
[EQUATION] Hence, for every [MATH] , we have [EQUATION] Finally, ( ), ( ) and ( ) imply that [MATH] , as desired. Thus, by the principle of mathematical induction, we have shown that ( ) holds for each [MATH] and each [MATH] . Now, let [MATH] be the unique node in [MATH] . Note that [MATH] . Hence, [MATH] . Therefore, ... |
Now, the first three items of Theorem are direct consequences of Lemma 4.1 . The super amalgamation property follows from Lemma 4.1 together with 21 , Theorem 5.3.5] |
Proof of Theorem (1) To prove the finite schema axiomatizability, it is enough to prove that [MATH] . Let [MATH] be any set of variables (finite or infinite) and let [MATH] . Then, |
[EQUATION] The implication [MATH] follows from the fact that [MATH] . For the other direction, suppose that [MATH] . Let [MATH] and let [MATH] be a finite set such that [MATH] . We can assume that [MATH] . By Theorem 3.3 iii ), there are [MATH] and [MATH] such that [MATH] and [MATH] . The fact that [MATH] implies that ... |
Now, we need to show that [MATH] . Let [MATH] . By the universal mapping property, there is an onto homomorphism [MATH] . In other words, [MATH] is a homomorphic image of [MATH] . Thus, [MATH] is a homomorphic image of [MATH] too. But [MATH] is a variety, so it contains all its free algebras and it is closed under [MAT... |
(2) The finite base property follows immediately from Lemma 4.1 , Theorem 3.3 and (1). (3) It is known that the finite schema axiomatizability and the finite base property imply the decidability of the equational theory, see, e.g., |
(4) All the algebras constructed in this section are locally finite dimensional. Thus [MATH] is (5) We showed that [MATH] is characterized by positive equations, so it is canonical variety. Thus, super amalgamation property follows from 21 , Theorem 5.3.5] |
5. Free algebras: atoms and zero-dimensional elements In this section we give the proof of Theorem . We start with the following. |
Theorem 5.1 The free algebra [MATH] is a two elements algebra, hence it is atomic. Proof. Straightforward since any finite Boolean algebra is atomic. |
Theorem 5.2 Let [MATH] be an infinite set, the free algebra [MATH] is atomless. Proof. (Essentially due to D. Pigozzi , 2.5.13] Let [MATH] be such that [MATH] . We show that [MATH] is not an atom in [MATH] . Note that there is a finite [MATH] such that [MATH] . Let [MATH] and let [MATH] be such that [MATH] . By the uni... |
To prove the remaining part of Theorem , we need to prove the following lemma. Let [MATH] and [MATH] be finite ordinals such that [MATH] , and let [MATH] be a non-empty finite set. Let [MATH] . Recall the unit [MATH] constructed in the previous section that witnesses the satisfiability of [MATH] |
Lemma 5.3 Suppose that [MATH] is even. There is a sequence [MATH] such that: (1) For every [MATH] [MATH] . In particular, [MATH] |
(2) For every [MATH] : If [MATH] is odd then [MATH] . If [MATH] is even then [MATH] Proof. Let [MATH] be the only element in [MATH] . If [MATH] , then we are done. So let us suppose [MATH] . Now since [MATH] then there exists a normal form [MATH] such that [MATH] By axiom (Ax 1b), we have [MATH] . Hence, [MATH] . Let [... |
Theorem 5.4 Let [MATH] be a non-empty finite set, the free algebra [MATH] is atomless. Proof. Let [MATH] be such that [MATH] . We need to show that [MATH] is not an atom in [MATH] . By Theorem 3.3 and Theorem (1), there are finite [MATH] [MATH] ), finite [MATH] and non-empty finite set [MATH] such that [MATH] Thus, the... |
Step 1: Given the normal form [MATH] , construct the unit [MATH] , the actual base [MATH] , the tail [MATH] and the evaluation [MATH] as constructed in the previous section. Recall that we have |
[EQUATION] Step 2: Let [MATH] be the sequence given in Lemma 5.3 . Extend [MATH] to [MATH] as follows: Choose a brand new element [MATH] and let [MATH] . Recall that [MATH] is a non-empty set of free generators, so one can find a normal form [MATH] such that [MATH] . Let [MATH] . Define the evaluation [MATH] as follows... |
[EQUATION] By a similar argument to the proof of Lemma 4.1 , one can see that [EQUATION] Step 3: Recall the sequence [MATH] . Let [MATH] . Recall that [MATH] is a partition of the unit, see Theorem 3.3 ), ( ii ). Then there exists a unique normal form [MATH] such that [MATH] . Similarly, there exists a unique normal fo... |
[EQUATION] Step 4: Now, we prove the following: For every [MATH] [EQUATION] We use induction on [MATH] . Since [MATH] and [MATH] , then [MATH] and [MATH] . Thus, by definition of normal forms, [MATH] and [MATH] . Hence, [MATH] . The induction step goes in a similar way. Suppose that [MATH] , for some [MATH] . Let [MATH... |
[MATH] . Remember [MATH] and [MATH] . By the induction hypothesis, without loss of generality, we may assume that [MATH] Recall the construction of the unit [MATH] . Note that [MATH] , and in fact |
[EQUATION] Remember that the labels of [MATH] ’s were distinct normal forms in [MATH] . Thus, by Theorem 3.3 ii ), we have the following. For each element [MATH] |
[EQUATION] Therefore, by ( 11 ), ( 14 ) and the assumption that [MATH] , we have [EQUATION] Remember that [MATH] and [MATH] were chosen such that [MATH] and [MATH] . Hence, by the induction hypothesis, |
[EQUATION] We also note that [MATH] and [MATH] . Thus, by ( 13 ), ( 15 ) and ( 16 ), [EQUATION] Therefore, by construction of normal forms, [MATH] and [MATH] . In other words, [MATH] . Hence, ( 12 ) follows by the principle of mathematical induction. |
In particular, there are two forms [MATH] each of which is satisfiable form below [MATH] inside the free algebra [MATH] 11 ). We also proved that these forms are disjoint ( 12 ). Therefore, [MATH] is not an atom in the free algebra [MATH] as desired. |
Zero dimensional elements in the free algebras Definition 5.5 Let [MATH] and let [MATH] . Define [MATH] , the dimension set of [MATH] . The element [MATH] is said to be zero-dimensional if and only if [MATH] |
Proof of Theorem The free algebra [MATH] contains only two elements [MATH] and [MATH] . Now, suppose that [MATH] . Let [MATH] be such that [MATH] and [MATH] . Then there are finite [MATH] and finite [MATH] such that [MATH] [MATH] and [MATH] . Thus, by Theorem 3.3 (iii), one can find [MATH] and two normal forms [MATH] s... |
Let [MATH] and [MATH] be the two units (defined in the previous section) witnessing the satisfiability of [MATH] and [MATH] respectively. We can suppose that [MATH] and [MATH] share the same tail [MATH] , while their actual bases are disjoint. Let [MATH] and [MATH] be the sequences given by Lemma 5.3 Suppose that [MATH... |
Recall the labels of the elements of [MATH] and [MATH] . For each [MATH] , let [MATH] . For each [MATH] , let [MATH] . By a similar argument to Lemma 4.1 , one can verify that |
[EQUATION] Thus, [MATH] and [MATH] Moreover, it is easy to see that there are [MATH] and [MATH] such that [MATH] . Hence, [EQUATION] |
Therefore, [MATH] is not zero-dimensional in the free algebra [MATH] . Otherwise, if [MATH] is zero-dimensional then [MATH] , which contradicts ( 19 ). |
6. Application in logic and related developments One way of having nice versions of first order logic is to keep the set of formulas as it is but consider generalized models when giving meaning for these formulas. Such a move was first taken by L. Henkin in |
. The general assignment models for first order logic, where the set of assignments of variables into a model is allowed to be an arbitrary subset of the usual one, was introduced by I. Németi |
. With selecting a subset of assignments, dependence between variables can be introduced into semantics. For a survey on generalized semantics, see |
For now, let us suppose that [MATH] is finite. By a suitable language we mean a set of [MATH] -many individual variables together with a set of relation symbols each of which is assigned a positive rank. For simplicity, we assume that our suitable languages do not contain functional symbols and/or constant symbols. Giv... |
Definition 6.1 Suppose that [MATH] is a suitable language. A general assignment model is an ordered pair [MATH] with [MATH] a standard first order model with domain [MATH] and interpretation function [MATH] , and [MATH] is a non-empty set of assignments on [MATH] , i.e.,, a subset of [MATH] , where [MATH] is the set of... |
[EQUATION] Here, [MATH] is the relation between assignments of identity up to [MATH] -values. We denote the logical system consists of the set of formulas in suitable language [MATH] together with the general assignment models by [MATH] . The notions of satisfiable formulas contradictions valid formulas , etc, are defi... |
Theorem 6.2 Let [MATH] be a suitable language. Each of the following is true. (1) [MATH] is finitely-schema axiomatizable. (2) [MATH] has the finite model property, i.e.,, every non-valid formula is falsified in a finite general assignment model. |
(3) The set of validities of [MATH] is decidable. (4) [MATH] has most of the positive definability properties: Craig’s interpolation, Beth definability, etc. |
(5) If [MATH] has at least one relation symbol, then every finitely axiomatizable theory in [MATH] cannot be both complete and consistent. |
Let [MATH] denotes Lindenbaum-Tarski algebra of [MATH] . Let [MATH] be the set of all atomic formulas in language [MATH] . To deduce the above theorem from our algebraic results herein, it would be enough to prove that [MATH] . The function assigning [MATH] , the meaning of [MATH] in [MATH] , to [MATH] is a homomorphis... |
We also note that a completely mechanical translation of the proofs of our algebraic theorems can be used to prove the above theorem. Such translation, from algebra to logic, was used in 42 , Chapter 2] to obtain the following results for guarded fragments of first order logic: (1) Every satisfiable formula of guarded ... |
6.1. Atomicity of free algebras in algebraic logic It was mentioned in the introduction that I. Németi used a metalogical proof (translation of Gödel’s incompleteness theorem) to show non-atomicity of finitely generated free algebras of [MATH] , if [MATH] . Such metalogical argument could be used also to deduce non-ato... |
and So far, only one atomicity result has been obtained. The proof that [MATH] is atomic, for finite [MATH] , relies on the facts that [MATH] is a discriminator variety and the equational theory of [MATH] coincides with the equational theory of the finite [MATH] ’s, c.f. , 2.5.7] . This could be generalized by H. André... |
as follows: For any variety of Boolean algebras with operators [MATH] of finite similarity type, if [MATH] is [EQUATION] In the literature of algebraic logic, there are several varieties (of finite similarity types) that are [MATH] and its variations [MATH] and [MATH] . The classes of non-commutative cylindric algebras... |
and . The finite algebra property of these classes can be found in and The question whether the finitely generated free algebras of these classes are atomic remained open for three decades. Recently, negative answers have been obtained for this problem. For finite [MATH] , the free algebras [MATH] [MATH] and [MATH] wer... |
. Non-atomicity of free non-commutative cylindric algebras [MATH] and [MATH] was proved in . In , similar non-atomicity result for the class [MATH] was obtained. Finally, preprint |
includes an idea for showing non-atomicity of [MATH] , for finite [MATH] . It is worthy of note that the methods in these references are quite different, in each case there is a different difficulty. |
Hence, roughly speaking, we can say that being non-discriminator in the above classes was more dominant than the finite algebra property, and it caused the non-atomicity of finitely generated free algebras. A natural question arises here: is that always true? |
Problem Find a variety of Boolean algebras with operators [MATH] such that: (a) the similarity type of [MATH] is finite, (b) [MATH] is |
(b) [MATH] is not discriminator variety, and (d) all the finitely generated free algebras of [MATH] are atomic. 6.2. Infinite dimensional free algebras |
Here, assume that [MATH] is an infinite ordinal. The free algebras of infinite dimensional cylindric algebras are more interesting. The metalogical technique used in |
provides a proof for non-atomicity of the free algebras of [MATH] . The non-atomicity of free algebras of [MATH] can be obtained by a purely algebraic argument, see |
. This argument uses the fact that [MATH] is generated as a variety by its locally finite dimensional algebras. The same is not true for classes [MATH] [MATH] and [MATH] , however with a different algebraic technique non-atomicity of the free algebras of these classes was shown in 42 , Appendix 2] |
We note that [MATH] is [MATH] cannot work here, it depends essentially on the existence of diagonals. The method used here to prove non-atomicity of the free algebras of [MATH] is completely different than the one in 42 , Appendix 2] Acknowledgment We are deeply indebted to the anonymous referee for his fruitful commen... |
# Source: arxiv 1807.00760 # Title: Role of thermal expansion heterogeneity in the cryogenic rejuvenation of metallic glasses # Sections: all # Downloaded: 2026-03-03T05:15:56.142479+00:00 |
Role of thermal expansion heterogeneity in the cryogenic rejuvenation of metallic glasses Abstract Cryogenic rejuvenation in metallic glasses reported in Ketov et al ’s experiment (Nature(2015)524,200) has attracted much attention, both in experiments and numerical studies. The atomic mechanism of rejuvenation has been... |
Introduction Glasses are solid materials, disordered at small scale, but homogeneous and isotropic at large scales. It has been understood for some time, however, that their elastic properties are not uniform if they are computed on mesoscale. This heterogeneous elasticity has been characterized by a number of simulati... |
Recently, it was discovered that an alternative to this mechanical rejuvenation could be obtained, in some metallic glasses (MGs), by thermal cycling towards low temperatures ketov2015rejuvenation . This is quite surprising, as from a purely thermal point of view one would expect that bringing the system to a lower tem... |
In this work we take an alternative route that bridges the gap between molecular dynamics and experiments, by using the molecular dynamics simulations to obtain quasi equilibrium properties and assessing the importance of the induced mechanical stresses by using a continuum description. Our first step is to quantify th... |
II The local thermal expansion coefficient The thermal expansion coefficient quantifies the volume strain caused by a change in temperature, and can be written for a macroscopic system as |
[EQUATION] where [MATH] is the volume strain induced by the temperature change [MATH] In metallic glasses, its order of magnitude is around [MATH] |
kato2008relationship In order to define a local thermal expansion coefficient, we use the coarse graining approach originally proposed by Goldhirsch and Goldenberg PhysRevE.80.026112 goldhirsch2002microscopic , that allows one to define continuum fields in the sense of hydrodynamics or elasticity, starting from atomic ... |
goldhirsch2002microscopic from the atomic positions: [EQUATION] where [MATH] is the displacement of atom [MATH] at time [MATH] , starting from a reference position, and [MATH] is the coarse graining function, in this paper we choose a Gaussian [MATH] , with [MATH] the coarse graining scale and [MATH] |
The local strain is calculated, under the assumption of small deformations, as [MATH] . We obtain the local thermal expansion coefficient (LTEC) from the local volumetric strain as |
[EQUATION] where [MATH] is the LTEC at position [MATH] , and [MATH] is local volumetric strain caused by the temperature change [MATH] |
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