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Universality on frames
Accepted Manuscript Universality on frames
P.S. Gevorgyan, S.D. Iliadis, Yu.V. Sadovnichy
PII: DOI: Reference:
S0166-8641(17)30053-6 http://dx.doi.org/10.1016/j.topol.2017.02.010 TOPOL 6018
To appear in:
Topology and its Applications
Received date: Revised date: Accepted date:
19 January 2016 2 July 2016 20 July 2016
Please cite this article in press as: P.S. Gevorgyan et al., Universality on frames, Topol. Appl. (2017), http://dx.doi.org/10.1016/j.topol.2017.02.010
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Universality on frames P.S. Gevorgyan1 , S.D. Iliadis2 , and Yu.V. Sadovnichy3
Abstract In the present paper we consider so-called saturated classes of frames of weight less than or equal to a given infinite cardinal τ . Such classes are saturated by universal elements. We consider three kinds of saturated classes. The saturated classes of any considered kind have the following basic property: the intersection of not more than τ many saturated classes is also a saturated class and therefore in this intersection there are also universal elements. We prove that the classes RegFrm(τ ) of regular and CRegFrm(τ ) of completely regular frames of weight ≤ τ are saturated classes in any of these kinds. This fact implies that in the classes RegFrm(τ, μ) and CRegFrm(τ, μ) of regular and completely regular frames, respectively, of weight ≤ τ and decomposition invariant (see [10]) μ ≤ τ there are universal elements. Key words: Lattice, Frame, Universal frame, Saturated class of frames, Decomposition invariant of frames. 2000 Mathematics Subject Classification: 06A06, 06D22, 54A05
1. Introduction A topological space T is said to be universal in a class Sp of topological spaces if (a) T ∈ Sp and (b) for each X ∈ Sp there exists a homeomorphism of X into T . If only the second condition is satisfied, then T is called a containing space for Sp. The problem whether there are universal elements actually can be posed for any class of spaces that is determined by a certain topological property. Such problems appeared in topology in its early development, when special classes of separable metrizable spaces were considered. With the consideration of more general classes of spaces, some methods of construction of universal elements appeared. These methods use factorization theorems, separating theorems and product of spaces. In [8] it is described a method of construction of universal and containing spaces in which does not use neither these theorems nor products of spaces. Let Sp be a class of spaces of weight less than or equal to a fixed infinite cardinal τ . The basic element of the method is the notion of a base B for a given collection S of elements of Sp (which is a fixed base of cardinality ≤ τ in each space of the collection) and a family R of equivalence relations on S. Using these two (fixed) elements, a space denoted by T(B, R) is constructed. This space is a containing space for the collection S of spaces. Suppose that for any collection S of elements of Sp some construction containing spaces T(B, R) belong to Sp. Then, T(B, R) will be a universal element in Sp. In the case where the class Sp is saturated by these kind of elements, this class is called saturated. 1
The simplicity of the elements of the above mentioned method of construction of universal elements give us the possibility to use a modification of this method in Lattice Theory and to apply the same scheme for the construction of universal frames. Let Fr be a class of frames. We will say that a frame T is universal in Fr if (a) T ∈ Fr and (b) for each F ∈ F there exists a homomorphism of T onto F . If only the condition (b) is satisfied, then T will be called a containing frame for Fr. Universal elements for different classes of frames (the class of all frames, the class of completely regular frames, the class of regular frames with countable bases, zero-dimensional frames (see [1]) of weight less than or equal to a fixed infinite cardinal are considered (in its localic version) in the paper [6] (see also related papers [11] and [12]). For the construction of universal elements in [5] for the class of all frames of a given infinite weight and in [9] for the classes of all regular frames and of all completely regular frames of any fixed infinite weight it is used a modification of the method of [8]. For the construction of classes of spaces having universal elements the different dimension-like invariants play an important role. Many such classes are defined as classes of spaces for which a given dimension-like invariant takes some concrete value. However, the dimension theory for frames is not sufficiently developed. Although for frames some dimension invariants are defined (see, for example, [2], [3], and [14] ) they do not define classes of frames with universal elements. In the paper [10] the notion of a saturated class of frames is introduced and a dimension-like invariant of a frame, the so-called decomposition invariant, is defined, which is a cardinal (in particular maybe a non-negative integer). It is proved that in the class Frm(τ, μ) of all frames of weight less than or equal to a fixed infinite cardinal τ having decomposition invariant ≤ μ ≤ τ , where μ is a fixed cardinal, is a saturated class and therefore in this class there are universal elements. In the present paper we consider saturated classes of frames of weight less than or equal to a given infinite cardinal τ . Such classes are saturated by universal elements. We consider three kind of saturated classes. The saturated classes of any considered kind have the following basic property: the intersection of not more than τ many saturated classes is also a saturated class and therefore in this intersection there are also universal elements. In Section 3 we prove that the class RegFrm(τ ) of regular and the class CRegFrm(τ ) of completely regular frames of weight ≤ τ are saturated classes in the sense of the paper [10]), which in the present paper are called saturated classes with ∧-mappings. By the basic property of saturated classes and since the above mentioned class Frm(τ, μ) is saturated, this fact implies that the classes RegFrm(τ, μ) and CRegFrm(τ, μ) of regular and completely regular frames, respectively, of weight ≤ τ and decomposition invariant μ ≤ τ , are also saturated classes and therefore in these classes there are universal elements. In Section 4 we introduce the notion of a saturated class of frames by a slightly different way from that of the paper [10]. This new kind of saturated classes, 2
which here are called saturated classes without ∧-mappings, have also the basic property of saturated classes. We prove that the classes RegFrm(τ ) and CRegFrm(τ ) are saturated in this new sense. We prove also that any saturated class without ∧-mappings is a saturated class with ∧-mappings. In Section 5, using the notions of τ -complete sets and its τ -closed subsets (see, for example, [4] or [7]), we introduce another kind of saturated classes having the above mentioned basic property of saturated classes and prove that the classes RegFrm(τ ) and CRegFrm(τ ) are also saturated in this new sense. Although the classes RegFrm(τ ) and CRegFrm(τ ) are saturated in any considered sense, in general it is an open problem if some of these kinds of saturated classes coincide.
2 Preliminaries 2.1 Notation. An ordinal number is considered as the set of smaller ordinal numbers, and a cardinal number is considered as the initial ordinal number of this cardinality. Therefore, for ordinals δ and τ relations δ ∈ τ + 1 and δ ≤ τ are equivalent. By τ we denote a fixed infinite cardinal. The set of all nonempty finite subsets of τ will be denoted by F. The symbol ≡ in a relation means that one or both sides of the relation are new notations. Each mapping f of the set τ onto a set B is called an indication of B and will be denoted by B = {aδ : δ ∈ τ } where aδ = f (δ), δ ∈ τ . Frames. Recall that a frame is a complete lattice L in which x∧ S = {x ∧ s : s ∈ S} for any x ∈ L and any S ⊆ L. Our notations shall be fairly standard from Picado and Pultr [13]. For instance we denote the top element and the bottom element of L by 1L and 0L respectively. For each x ∈ L we put x∗ = {z ∈ L : z ∧ x = 0L }. The correspondence x → x∗ will be called the star operator. For x, y ∈ L we write y ≺ x iff y ∗ ∨x = 1L and y ≺≺ x iff there is a system {cr ∈ L : r ∈ Q∩[0, 1]} (Q is the rational numbers) such that c0 = y, c1 = x and c r ≺ cs whenever r < s. A frame L is said to be regular if for each x ∈ L, x = {y ∈ L : y ≺ x} and completely regular if for each x ∈ L, x = {y ∈ L : y ≺≺ x}. A subset B of a complete lattice (or frame) L is called a base for L if each element of L is a supremum of a subset of B. The weight of a complete lattice (or frame) L is the minimal cardinal κ for which there exists a base B for L of cardinality κ. A frame homomorphism is a map between frames which preserves finite meets, including the top element, and arbitrary joins, including the bottom element.
3
Let L be a class of complete lattices (or frames). We say that an element T ∈ L is universal in this class if for every L ∈ L there exists a homomorphism of T onto L. 2.2 Definitions (see [10]). Let L be an indexed collection of frames (and, therefore, according to our agreements the weights of all elements of L are ≤ τ ). Indexed base for L. Suppose that for every L ∈ L we choose an indexed base of L: B L ≡ {aL δ : δ ∈ τ} L such that aL 0 = 0L and a1 = 1L . Then, the indexed set
B ≡ {B L ≡ {aL δ : δ ∈ τ } : L ∈ L}
(2.1)
will be called a base for L. If all B L , L ∈ L, are closed under taking finite infima and with respect to the star operator, then B is called also closed under taking finite infima and with respect to the star operator. Suppose that for each L ∈ L we have a mapping θ[L] of F into τ such that for L every δ ∈ τ and t ∈ F we have θ[L]({δ}) = δ and aL θ[L](t) = ∧{aδ : δ ∈ t}, then θ[L] will be called the ∧-mapping in L and the set θB ≡ {θ[L] ≡ θB [L] : L ∈ L}
(2.2)
will be called the set of ∧-mappings for B. We note that in general θ[L] is not uniquely determined. Extension of a base. Let B0 ≡ {B 0,L ≡ {a0,L : δ ∈ τ } : L ∈ L}, B1 ≡ {B 1,L ≡ {a1,L : δ ∈ τ } : L ∈ L} δ δ be two bases for L and ϕ a mapping of τ into itself such that ϕ(0) = 0 and ϕ(1) = 1. We say that B1 is an extension of B0 with extension mapping ϕ (from B0 to B1 ) if for every L ∈ L, 0,L a1,L ϕ(δ) = aδ , δ ∈ τ.
In general, the mapping ϕ is not uniquely determined. If the bases B0 and B1 are considered with the sets θB0 ≡ {θB0 [L] : L ∈ L} and θB1 ≡ {θB1 [L] : L ∈ L} of ∧-mappings respectively, then the base B1 is said to be a proper extension of B0 if for every t ∈ F and L ∈ L we have θB1 [L](ϕ(t)) = ϕ(θB0 [L](t)). It is easy to prove that if B1 is a (proper) extension of B0 and a base B2 for L is a (proper) extension of B1 , then B2 is also a (proper) extension of B0 . 4
Moreover, if ϕ0 is a (proper) extension mapping from B0 to B1 and ϕ1 is a (proper) extension mapping from B1 to B2 , then ϕ1 ◦ ϕ0 is a (proper) extension mapping from B0 to B2 . Admissible family of equivalence relations. A family R ≡ {∼s : s ∈ F}
(2.3)
of equivalence relations on L is said to be admissible if: (a) whenever ∅ = p ⊂ s ∈ F we have ∼s ⊂ ∼p and (b) for any s ∈ F the number of equivalence classes of the relation ∼s is less than or equal to τ . We denote by C(∼s ) the set of all equivalence classes of the relation ∼s and put C(R) = ∪{C(∼s ) : s ∈ F}. Note that the cardinality of C(R) is ≤ τ . Final refinement. Let R0 ≡ {∼s0 : s ∈ F} and R1 ≡ {∼s1 : s ∈ F} be two admissible families of equivalence relations on L. It is said that R1 is a final refinement of R0 if for each s ∈ F there exists t ∈ F such that ∼t1 ⊂ ∼s0 . Obviously, if an admissible family R2 is a final refinement of R1 and R1 is a final refinement of R0 , then R2 is a final refinement of the family R0 . The family RB of equivalence relations on L. Let (2.1) be a base for L closed under taking finite infima and considered with the set (2.2) of ∧mappings. We denote by RB ≡ {∼sB : s ∈ F} the family of equivalence relations on L defined as follows: two elements L and L of L are ∼sB -equivalent for some s ∈ F if and only if for each δ, η ∈ s and L L L t ⊂ s we have (a) the relation aL δ ≤ aη implies the relation aδ ≤ aη and (b) θ[L](t) = θ[L ](t). B-admissible family. Let (2.1) be a base for L closed under taking finite infima and considered with the set (2.2) of ∧-mappings. An admissible familly (2.3) of equivalence relations on L is said to be B-admissible if this family is a final refinement of RB . It is easy to see that if R0 is a B-admissible family of equivalence relations on L and R1 is an admissible family, which is a final refinement of R0 , then R1 is also B-admissible. The Containing Frame T(B, R). For any collection L of mutually disjoint frames, an arbitrary base B for L considered with the set of ∧-mappings, and an arbitrary B-admissible family R of equivalence relations on L, the construction of the Containing Frame T(B, R) is given in [10]. In [5] it is given the same construction of T(B, R) considered that R coincides with the family RB . We recall this construction. 5
Let L be a collection of mutually disjoint frames and U the (free) union of elements of L. Further, let (2.1) be a base for L, closed under taking finite infima, and (2.3) a B-admissible family of equivalence relations on L. For every δ ∈ s ∈ F and H ∈ C(∼s ) we put L Aδ (H) = {aL δ : L ∈ H} ∪ {a0 = 0L : L ∈ L \ H} ⊂ U
and
BT = {Aδ (H) : δ ∈ s ∈ F, H ∈ C(∼s )}.
For every L ∈ L and a subset M ≡ {Aδj (Hj ) : j ∈ J} ⊂ BT we put and
L ML = {aL δ ∈ L : {aδ } = Aδ (H) ∩ L, Aδ (H) ∈ M }
= {sup ML : L ∈ L}. M
Now we consider the (non-indexed) set : M ⊆ B}. T ≡ T(B, R) ≡ {M ≤N for M , N ∈ T if and only if for In T we define an order ≤ by putting M every L ∈ L we have sup ML ≤ sup NL . Obviously, the pair (T, ≤) is a poset. ∈ T. The generators of the The subset M ⊂ BT will be called a generator of M elements of T are not uniquely determined. It is easy to see that the union of any number of generators of an element of T is also a generator of this element. Therefore, the union of all generators of a certain element of T is the maximal generator of that element. 2.3 Theorem. (See [10].) The above constructed poset T ≡ T(B, R) is a frame, the set BT is a base of T, and for each L ∈ L the mapping hL : T → L ) = sup ML , M ∈ T, is an onto homomorphism. defined by relation hL (M
3 On regular and completely regular frames Definition of a saturated class of frames. We recall the definition of a saturated class of frames given in [10]. A class Fr of frames is said to be saturated (with respect to the universality) if for any indexed collection L of mutualy disjoint elements of Fr there exists a base B0 (considered with a set of ∧-mappings) for L having the property: for each proper extension B of B0 there exists a Badmissible family RB of equivalence relations on L such that for each admissible family R of equivalence relations on L, which is a final refinement of RB , the Containing Frame T(B, R) (constructed in [10] and presented above) belongs to Fr. The base B0 is called an initial base corresponding to the collection L (and
6
the class Fr) and the family RB is called an initial family corresponding to the base B, the collection L, (and the class Fr). 3.1 Theorem The class RegFrm(τ ) of all regular frames of weight ≤ τ is saturated. Proof. We shall use the notation considered in the proof of Theorem 3.1 of [9]. In particular L is a collection of regular mutually disjoint frames of RegFrm(τ ) such that each element L ∈ RegFrm(τ ) is isomorphic to an element of L and U is the union of elements of L. Also, for each L ∈ L, B L is a regular base of L of cardinality less than or equal to τ closed under taking finite infima and the B L -star operator and (3.1) B L = {aL δ : δ ∈ τ} is a special indication of B L constructed in [9] (see relation (3.1) of [9]). We put B ≡ {{ aL δ : δ ∈ τ } : L ∈ L} and as the set θB = {θL : L ∈ L} of ∧-mappings for B we take the set for which θL = θ for each L ∈ L, where θ is the mapping of F into τ constructed in Theorem 3.1 of [9]. We prove that B is an initial base for L. Indeed, let B be a proper extension of B. We prove that the family RB is an initial family of L corresponding to the base B and the collection L proving simultaneously that B is an initial base for L. It suffices to prove that the Containing Frame T(B , R) is regular, for each admissible family R of equivalence relations on L, which is a final refinement of RB . Let R ≡ {∼s : s ∈ F} be an admissible family of equivalence relations on L, which is a final refinement of RB . By Proposition 4.3 of [10] we have T(B, R) = T(B , R). Therefore, it suffices to prove that the Containing Frame T(B, R) is regular. Recall that the set BT = {Aδ (H) : δ ∈ s ∈ F, H ∈ C(∼s )}, where
Aδ (H) = {aL δ : L ∈ H} ∪ {0L : L ∈ L \ H},
is a base for T ≡ T(B, R). We also consider the Containing Frame T(B, RB ) constructed in [9]. The set {Aδ : δ ∈ τ }, where Aδ = {aL δ : L ∈ L}, is a base for this frame. Here we shall denote this base by B(T). (In [9] this base was denoted by BT .) Since for each δ ∈ τ and any s ∈ F we have Aδ = {Aδ (H) : H ∈ C(∼s )}, each element of B(T) and therefore each element of T(B, RB ) is an element of T. It is clear that the order on T(B, RB ) coincides with the order induced by 7
T. Moreover, the top and the bottom elements of T coincide with the top and bottom elements of T(B, RB ) and will be denoted by 1T and 0T respectively. For each A ∈ B(T) the pseudocomplement of A in T(B, RB ) will be denoted by A∗ (not to be confused with its pseudocomplement A∗ in T). We note that A∗ ≤ A∗ . For each A ≡ {aL A : L ∈ L} and H ∈ C(R) we put A0 (H) = {aL A : L ∈ H} ∪ {0L : L ∈ L \ H}. It is easy to see that A0 (H) is an element of T. Indeed, if A = {Aδ ∈ B(T) : Aδ ∈ M ⊂ B(T)}, then
A0 (H) =
{(Aδ )0 (H) = Aδ (H) ∈ BT : Aδ ∈ M } ∈ T.
To prove the theorem it suffices to prove that for each δ ∈ s ∈ F and any H ∈ C(∼s ) we have (3.2) Aδ (H) = {B ∈ T : Aδ (H) ∨ B ∗ = 1T }. Let δ ∈ s ∈ F and H ∈ C(∼s ). Since the frame T(B, RB ) is regular we have (3.3) Aδ = {A ∈ T(B, RB ) : Aδ ∨ A∗ = 1T }. Obviously, Aδ (H) ∧ Aδ = Aδ (H). Therefore, Aδ (H) = Aδ (H) ∧ Aδ = Aδ (H) ( {A : Aδ ∨ A∗ = 1T } = =
{Aδ (H) ∧ A : Aδ ∨ A∗ = 1T }.
(3.4)
Let A ≡ {aL A : L ∈ L} be an element included in the relation (3.3), Then A ≤ Aδ and therefore (3.5) Aδ (H) ∧ A = A0 (H). Putting
A10 (H) = {0T : L ∈ H} ∪ {1L : L ∈ L \ H}
we have A10 (H) ∈ T and A10 (H) ∧ A0 (H) = 0T . The last relation implies that A10 (H) ≤ (A0 (H))∗ . Therefore, (A0 (H))∗ ∩ L = 1L , L ∈ L \ H.
(3.6)
0
On the other hand: (a) the relation A (H) ≤ A implies that A∗ ≤ A∗ ≤ (A0 (H))∗
(3.7)
L and (b) the relation Aδ ∨ A∗ = 1T implies that aL δ ∨ aA = 1T for each L ∈ L and in particular for each L ∈ H. Therefore, if (A0 (H))∗ = {bL : L ∈ L}, then by relation (3.7) for every L ∈ H, L aL δ ∨ b = 1T .
8
(3.8)
Relations (3.6) and (3.8) imply that Aδ (H) ∨ (A0 (H))∗ = 1T .
(3.9)
Thus, from relations (3.4), (3.5), and (3.9) it follows that Aδ (H) = {A0 (H) : Aδ ∨ A∗ = 1T } ≤ {A0 (H) : Aδ (H) ∨ (A0 (H))∗ = 1T } ≤ ≤
{B ∈ T : Aδ (H) ∨ B ∗ } ≤ Aδ (H)
proving relation (3.2) and completing the prove of the theorem. The proof of the following theorem is similar to the proof of Theorem 3.1. 3.2 Theorem. The class CRegFrm(τ ) of all completely regular frames of weight ≤ τ is saturated. We recall the notion of the decomposition invariant of a frame. Let L be a frame and μ ≤ τ a cardinal. We shall say that the decomposition invariant of L is less than or equal to μ and write dec(L) ≤ μ if there are: (a) zerodimensional frames F η , η ∈ μ + 1 (here, μ is considered as an initial ordinal), and (b) homomorphisms hη of L onto F η , η ∈ μ + 1, such that for each a, b ∈ L the condition hη (a) = hη (b) for every η ∈ μ + 1 implies that a = b. The following corollary follows from the above Theorems 3.1 and 3.2 and Theorems 3.9, 3.10, and 4.7 of [10]. 3.3 Corollary. The class RegFrm(τ, μ) (respectively, the class CRegFrm(τ, μ)) of all regular (respectively, of all completely regular) frames of weight ≤ τ with the decomposition invariant less than or equal to μ ≤ τ is saturated and therefore in this class there are universal elements.
4 On saturated classes Let L be a collection of mutually disjoint frames. In Section 2 for a given base B with a set θB of ∧-mapping we recalled the definition of the family RB of equivalence relations on L and the construction of the Containing Frame T(B, R) for each B-admissible family R of equivalence relations on L. In the present section, in order to distinguish notation, we shall denote such a base B by B[∧] and the family RB by RB[∧] ≡ {∼sB[L] : s ∈ F}. Then, the Containing Frame T(B, R) will be denoted by T(B[∧], R) and therefore the Containing Frame T(B, RB ) by T(B[∧], RB[∧] ). Let
B ≡ {{aL δ : δ ∈ τ } : L ∈ L}
9
(4.1)
be a base for L which is considered without any set of ∧-mappings. We define the family (4.2) RB ≡ {∼sB : s ∈ F} of equivalence relations on L as follows: two elements L and L of L are ∼sB equivalent for some s ∈ F if and only if for each η ∈ s and t ⊂ s the relation L {aL δ : δ ∈ t} = aη implies the relation
L {aL δ : δ ∈ t} = aη .
L L L Therefore, for δ ∈ s the relation aL δ ≤ aη implies the relation aδ ≤ aη .
As in Section 2 an admissible family R ≡ {∼s : s ∈ F}
(4.3)
of equivalence relations on L is said to be B-admissible if for each s ∈ F there exists t ∈ F such that ∼t ⊂ ∼sB . Let (4.1) be a base for L and (4.3) a B-admissible family of equivalence relations on L. Then, we can define a poset T(B, R) exactly as in Section 2 we define the Containing Frame T(B, R). (We note that in that definition we did not use the set of ∧-mappings for B. This set is used for the definition of the family RB , which determines what families R we can consider). Although Theorem 4.1 given below has the same formulation of Theorem 2.3 it does not follow from this Theorem because the families R in both theorems are distinct in general. 4.1 Theorem. The poset T ≡ T(B, R) is a frame, the set BT is a base of T, and ) = sup ML , for each L ∈ L the mapping hL : T → L defined by relation hL (M M ∈ T, is an onto homomorphism. Proof. The proof of the theorem is similar to that of Theorem 1.3 of [5]. It follows from the following three lemmas. Lemma 1. The poset (T, ≤) is a complete lattice of weight ≤ τ . Lemma 2. For every L ∈ L there exists a homomorphism hL of T onto L. Lemma 3. The complete lattice (T, ≤) is a frame. The proofs of Lemmas 1 and 3 are the same as the proofs of the corresponding lemmas of Theorem 1.3 of [5]. Although the proof of Lemma 2 follows the proof
10
of Lemma 2 of Theorem 1.3 we shall present here a part of this proof since some notions, which have the same notations, are different. The proof of Lemma 2. Let L be a fixed element of L. The definition of the mapping hL and the proofs of the facts that it is onto and preserves suprema are the same as in Lemma 2 of the above mentioned theorem. We shall prove that the mapping hL preserves finite infima. Let A1 ≡ Aδj (Hj ) and A2 ≡ Aηi (Hi ) be two fixed elements of BT such that δj ∈ sj , Hj ∈ R(∼sj ), ηi ∈ si , and Hi ∈ R(∼si ). We shall define a subset of BT , denoted by M (A1 , A2 ) ≡ Aδj (Hj )ΔAηi (Hi ) as follows. If the set Hj ∩ Hi is empty, then we put M (A1 , A2 ) = {0T }. If Hj ∩ Hi = ∅, then we put M (A1 , A2 ) = {Aδ (H) : {δ} ∪ sj ∪ si ⊂ s ∈ F, H ∈ C(∼s ),
L L H ⊂ Hj ∩ Hi , aL δj ∧ aδi = aδ for each L ∈ H}
It is clear that for every Aδ (H) ∈ M (A1 , A2 ) we have Aδ (H) ≤ Aδj (Hj ) and Aδ (H) ≤ Aηi (Hi ). Moreover, it is easy to verify that if δ ≡ δj = δi and Hj = Hi , then Aδj (Hj ) = Aηi (Hi ) ≤ {Aδ (H ) ∈ M (A1 , A2 ) : H ∈ C(∼sj ∪si )}.
(4.4)
(4.5)
and K be two elements of T. We need to prove that Now, let N ∧ K) = sup NL ∧ sup KL . hL (N
(4.6)
Suppose that N = {Aδj (Hj ) : j ∈ J, Hj ∈ C(∼sj ), δj ∈ sj ∈ F} and
K = {Aηi (Hi ) : i ∈ I, Hi ∈ C(∼si ), ηi ∈ si ∈ F}.
Without loss of generality we can suppose that N and K are the maximal and K, respectively. generators of N Consider the set M≡
{Aδj (Hj )ΔAηi (Hi ) : (j, i) ∈ J × I}
11
. First, we shall prove that and denote by Mmax the maximal generator of M =N ∧ K. M
(4.7)
Obviously, for every (j, i) ∈ J × I we have and Aη (Hi ) ≤ K. Aδj (Hj ) ≤ N i
(4.8)
Then, since N and K are maximal generators, by the definition of the set M and the relations (4.4) and (4.8), for every (j, i) ∈ J × I we have Aδj (Hj )ΔAηi (Hi ) ⊂ N ∩ K and, therefore, M ⊂ N and M ⊂ K. This means that sup ML ≤ sup NL and sup ML ≤ sup KL .
(4.9)
Relation (4.9) implies that ≤N and M ≤ K. M Now suppose that for an element P ∈ T we have and P ≤ K. P ≤ N Since N and K are maximal generators we have P ⊂ N and P ⊂ K. Let Aδ (H) be an element of P . Then, Aδ (H) ∈ N ∩K. This means that for some j ∈ J we have δ = δj , H = Hj ∈ C(∼sj ), and therefore A1 ≡ Aδ (H) = Aδj (Hj ). Similarly, for some i ∈ I we have δ = ηi , H = Hi ∈ C(∼si ), and therefore A2 ≡ Aδ (H) = Aηi (Hi ). In this case, all Aδ (H ) included in formula (4.5), as elements of M (A1 , A2 ), are elements of M . This means that Aδ (H) = Aδj (Hj ) . Thus, P ⊂ Mmax and therefore is an element of the maximal generator of M completing the proof of the relation (4.7). P ≤ M We shall now prove that sup ML = sup NL ∧ sup KL .
(4.10)
(Recall that L is a fixed element of L.) First we shall prove that sup NL ∧ sup KL ≤ sup ML .
(4.11)
Let b ∈ NL , c ∈ KL , and a = b ∧ c. It suffices to prove that a ∈ ML . This is clear if a = 0L . Suppose that a = 0L . There exist A1 ≡ Aδj (Hj ) ∈ N with Hj ∈ C(∼sj ) and A2 ≡ Aηi (Hi ) ∈ K with Hi ∈ C(∼si ) such that b = aL δj , c = L L aηi , and L ∈ Hj ∩ Hi . There exists δ ∈ τ such that a = aδ . Let s = {δ} ∪ sj ∪ si 12
and H be the element of C(∼s ) containing L. Then, Aδ (H) ∈ M (A1 , A2 ) ⊆ M which means that a = aL δ ∈ ML proving relation (4.11). Now we prove that sup ML ≤ sup NL ∧ sup KL .
(4.12)
Let a ∈ ML . It suffices to find b ∈ NL and c ∈ KL such that a ≤ b ∧ c. For a = 0L we can take b = c = 0L . Therefore we can suppose that a = 0L . There exist δj ∈ sj ∈ F, Hj ∈ C(∼sj ), ηi ∈ si ∈ F and Hi ∈ C(∼si ) such that for some δ ∈ τ and H ∈ C(∼s ), where s = {δ} ∪ sj ∪ si , we have L ∈ H, a = aL δ , and Aδ (H) ∈ Aδj (Hj )ΔAηi (Hi ). L L L The last relation means that aL δ = aδj ∧ aηi . Therefore, we can put b = aδj and c = aL ηi completing the proof of relation (4.12) and therefore relation (4.10). Relations (4.7) and (4.10) imply relation (4.6) completing the proofs of the lemma and theorem.
The following proposition follows immediately from the construction of Containing Frames T(B, R) and T(B[∧], R). 4.2 Proposition. Suppose that the base B is considered with a set of ∧mappings and the family (4.3) is B-admissible and B[∧]-admissible. Then, the Containing Frames T(B, R) and T(B[∧], R) coincide. Saturated classes without ∧-mappings. Now we give a notion of a saturated class using bases for a collection of frames without the sets of ∧-mappings. A class Fr of frames is said to be saturated if for every collection L of elements of Fr there exists a base B0 for L satisfying the following property: for each extension B of B0 there exists a B-admissible family RB of equivalence relations on L such that for each admissible family R of equivalence relations on L, which is a final refinement of RB , the Containing Frame T(B, R) belongs to Fr. In this case the base B0 is called an initial base corresponding to the collection L of elements of Fr and the family RB will be called an initial family corresponding to the base B and the collection L. Saturated classes in the above sense will be called saturated classes without ∧-mappings and saturated classes considered in Section 2 will be called saturated classes with ∧-mappings. 4.3 Theorem. Any saturated class of frames without ∧-mappings is a saturated class with ∧-mappings. Proof. Let Fr be a saturated class of frames without ∧-mappings. Consider an arbitrary collection L of elements of Fr and let B0 be an initial base corresponding to the collection L. Now, consider an arbitrary set of ∧-mappings for the base B0 . We shall prove that the base B0 [∧], that is, the base B0 considered 13
with the set of ∧-mappings, is an initial base (with respect to saturated classes with ∧-mapings) for the collection L. Indeed, let B[∧] be a proper extension of B0 [∧]. Then, the base B is an extension of B0 . Therefore, there exists a B-admissible family RB which is initial corresponding to the base B and the collection L. Denote by RB[∧] an admissible family of equivalence relations on L, which is simultaneously a final refinement of the families RB and RB[∧] . We prove that RB[∧] is initial corresponding to the base B[∧] and the collection L proving simultaneously that B[∧] is an initial base. Indeed, let R be an admissible family of equivalence relations on L, which is a final refinement of RB[∧] . Then, R is a final refinement of RB , which means that the Containing Frame T(B, R) belongs to Fr. By Proposition 4.2, T(B, R) = T(B[∧], R) and therefore T(B[∧], R) ∈ Fr proving that RB[∧] is initial corresponding to the base B[∧] and completing the proof of the theorem. The proof of the next theorem is similar to that of Theorem 3.1. We note that in Theorem 3.1 the set of ∧-mappings is use only for the construction of the initial base for L. 4.4 Theorem. The classes of regular and completely regular frames of weight ≤ τ are saturated classes without ∧-mappings. The proof of the following theorem is the same as the one of Theorem 3.10 of [10] 4.5 Theorem. The intersection of not more than τ many saturated classes without ∧-mappings is also a saturated class without ∧-mappings.
5. On a new type of saturated classes of frames τ -Complete sets (see, for example, [4] or [7]). Let C be a directed set (that is, for each c1 , c2 ∈ C there exists c ∈ C such that c1 ≤ c and c2 ≤ c). For a subset K ⊂ C we denote by sup(K) an element c ∈ C (if such an element exists) with the properties: (a) k ≤ c for each k ∈ K and (b) if c is an element of C such that k ≤ c for each k ∈ K, then c ≤ c. Obviously, if there exists sup(K), then it is uniquely determined. The set C is said to be τ -complete if for each chain K ⊂ C (that is, K is a linearly ordered subset) of cardinality ≤ τ there exists sup(K) ∈ C. A subset D of C is said to be τ -closed if for each chain K ⊂ D of cardinality ≤ τ we have sup(K) ∈ D. A subset K is said to be cofinal if for each e ∈ C there exists k ∈ K such that e ≤ k. We recall the following lemma. 5.1 Lemma. The intersection of not more than τ many non-empty τ -closed cofinal subsets of a τ -complete set C is also a non-empty τ -closed cofinal subset of C. 14
Agreement. In this section we suppose that L is a collection of mutually disjoint frames of weight ≤ τ . It is also assumed that any considered base B for L has cardinality ≤ τ and closed under taking finite infima. All bases for L will be considered without the set of ∧-mappings. The sets P ≡ P(L) and P ≡ P(L). We denote by P ≡ P(L) the set of all pairs (B, R), where B is a base for L and R is a B-admissible family of equivalence relations on L. On this set we a define a preorder ≤ setting that (B1 , R1 ) ≤ (B2 , R2 ) iff B2 is an extension of B1 and R2 is a final refinement of R1 . This preorder defines on L the natural equivalence relation, denoted by ∼, that is (B1 , R1 ) ∼ (B2 , R2 ) iff (B1 , R1 ) ≤ (B2 , R2 ) and (B2 , R2 ) ≤ (B1 , R1 ). The equivalence classes of the relations ∼ will be denoted by P ≡ P(L). The equivalence class containing an element (B, R) ∈ P will be denoted by [(B, R)]. On the set P we define an order ≤ setting [(B1 , R1 )] ≤ [(B2 , R2 )] iff (B1 , R1 ) ≤ (B2 , R2 ). It is easy to prove that the order ≤ on P is well defined, and that is it is independent of the pairs (B1 , R1 ) and (B2 , R2 ) of elements [(B1 , R1 )] and [(B2 , R2 )] respectively. 5.3 Proposition. The set P is τ -complete. Proof. It is easy to see that P is directed. Let K be a subset of P of cardinality ≤ τ (in particular, K may be a chain). We must prove that there is sup(K) ∈ P. In particular, this fact will imply that P is directed. Consider an arbitrary indication of K: K ≡ {[(Bδ , Rδ )] : δ ∈ τ }, where and
Bδ ≡ {{aL δ,ε : ε ∈ τ } : L ∈ L} Rδ ≡ {∼sδ : s ∈ F}.
Let χ : τ × τ → τ be a bijection. We define a base B ≡ {{aL η : η ∈ τ } : L ∈ L} L −1 (η). Therefore, the mapping ϕδ : for L setting aL η = aδ,ε , where (δ, ε) = χ τ → τ for which ϕδ (ε) = χ(δ, ε) , ε ∈ τ , is an extension mapping from Bδ to B. Also, we define an admissible family
R ≡ {∼s : s ∈ F} of equivalence relations on L by setting ∼s = ∩{∼sδ : δ ∈ s}. We prove that the pair (B, R) is an element of P and [(B, R)] = sup(K).
15
First we prove that (B, R) ∈ P, that is R is B-admissible. Let s ∈ F. We must prove that there exists t ∈ F such that ∼t ⊂ ∼sB . Let s ≡ {η1 , ..., ηn } ∈ F, i = 1, ..., n, and
χ−1 (ηi ) = (δi , εi ), i = 1, ..., n.
Since P is directed there exists δ ∈ τ such that [(Bδi , Rδi )] ≤ [(Bδ , Rδ )], i = 1, ..., n. Let ϕi be an extension mapping from Bδi to Bδ and ϕi (δi , εi ) = (δ , εi ), i = i, ..., n. We put Then,
s ≡ {εi : i = 1, ..., n} ∈ F. L L aL ηi = aδi ,εi = aδ ,εi , i = 1, ..., n, L ∈ L.
The last relation implies that ∼sB = ∼sBδ . Since R is a final refinement of Rδ
and Rδ is a final refinement of RBδ there exists t ∈ F such that ∼t ⊂ ∼sBδ and therefore ∼t ⊂ ∼sB proving that R is a final refinement of RB .
Now, we prove that [(B, R)] = sup(K). Let (B∗ , R∗ ), where R∗ = {∼s∗ : s ∈ F}, be an element of P such that [(Bδ , Rδ )] ≤ [(B∗ , R∗ )], δ ∈ τ, that is B∗ is an extension of Bδ with an extension mapping ϕδ∗ and R∗ is a final refinement of Rδ . We must prove that [(B, R)] ≤ [(B∗ , R∗ )], that is B∗ is an extension of B and for each s ∈ F there exists t ∈ F such that ∼t∗ ⊂ ∼s . It is easy to see that the mapping ϕ∗ : τ → τ such that for each η ∈ τ , ϕ∗ (η) = ϕδ∗ (ε), where (δ, ε) = χ−1 (η), is an extension mapping from B to B∗ , that is B∗ is an extension of B. Let s ∈ F. By the definition of the equivalence relation ∼s we have ∼s = ∩{∼sδ : δ ∈ s}. For each δ ∈ s there exists tδ ∈ τ such that ∼t∗δ ⊂ ∼sδ . Since R∗ is admissible there exists t ∈ F such that ∼t∗ ⊂ ∼t∗δ for each δ ∈ s and therefore ∼t∗ ⊂ ∩{∼t∗δ : δ ∈ s} ⊂ ∼s completing the proof of the proposition.
16
5.4 Proposition. Let [(B1 , R1 )] = [(B2 , R2 )] ∈ P. Then, the Containing Frames T1 ≡ T(B1 , R1 ) and T2 ≡ T(B2 , R2 ) coincide. Proof. Since B2 is an extension of B1 by Lemma 4.3 of [10] it follows that T(B1 , R2 ) = T(B2 , R2 ). (We note that in Lemma 4.3 of [10] we actually did not use the set of ∧-mappings.) Therefore it suffices to prove that T(B1 , R1 ) = T(B1 , R2 ).
(5.1)
Let s s B1 ≡ {{aL δ : δ ∈ τ } : L ∈ L}, R1 ≡ {∼1 : s ∈ F}, and R2 ≡ {∼2 : s ∈ F}.
Consider the bases BT1 ≡ {Aδ (H) : δ ∈ s ∈ F, H ∈ C(∼s1 )} and
BT2 ≡ {Bδ (G) : δ ∈ s ∈ F, G ∈ C(∼s2 )}
of T1 and T2 , respectively, where Aδ (H) = {aL δ : L ∈ H} ∪ {0L : L ∈ L \ H} and
Bδ (G) = {aL δ : L ∈ G} ∪ {0L : L ∈ L \ G}.
Let Aδ (H) be an arbitrary fixed element of BT1 , where H ∈ C(∼s1 ). Since R2 is a final refinement of R1 there exists t ∈ F such that ∼t2 ⊂ ∼s1 . Therefore, H = ∪{Gi : i ∈ I} ⊂ C(∼t2 ). Then, Aδ (H) = {Aδ (Gi ) : i ∈ I} proving that T1 ⊂ T2 . Similarly, T2 ⊂ T1 . Therefore, T1 = T2 which completes the proof of the proposition. Thus, if e ≡ [(B, R)] ∈ P, then we can use the notation T(e) ≡ T(B, R). Saturated class of frames by τ -complete sets. A class Fr of frames is said to be saturated (by τ -complete sets) if for every indexed collection L of mutually disjoint elements of Fr the set of all elements e ∈ P(L) for which T(e) ∈ Fr contains a τ -closed cofinal subset of P(L). 5.5 Theorem. The classes of regular and completely regular frames of weight ≤ τ are saturated by τ -complete sets. Proof. Consider the class RegFrm(τ ) and let L ba a collection of elements of RegFrm(τ ). In Theorem 3.1 we proved that there exists a base B for L such that for each extension B of B the Containing Space T(B , RB ) is an element 17
of RegFrm(τ ). This means that setting e ≡ [(B , RB )] and e = [(B, RB )] we have e ≤ e and T(e ) ∈ RegFrm(τ ). Then, the proof of the theorem follows from the fact that the subset {e ∈ P(L) : e ≤ e } is τ -closed. The following theorem follows from Lemma 5.1. 5.6 Theorem. The intersection of not than τ many saturated classes of frames by τ -complete sets is also a saturated class by τ -complete sets and therefore in this intersection there are universal elements. 5.7 Problem. Do some of the considered kinds of saturated classes coincide?
References [1] B. Banaschewski, Universal zero-dimensional compactifications, Proc. Int. Conf. on Categorical Topology, Prague 1988, World Sci. Pub. Singapore, 1989, 257 − 269. [2] M.G. Charalambous, Dimension theory for σ-frames, J. London Math. Soc. (2), 8(1974), 149-160. [3] M.G. Charalambous, A new covering dimension function for uniform spaces, J. London Math. Soc. (2), 11 (1975) 137-143. [4] A. Chigogidze, Inverse Spectra, North-Holland Mathematical Library, Vol. 53, North-Holland Publishing Co., Amsterdam, 1996. [5] Themba Dube, Stavros Iliadis, Jan van Mill, and Inderasan Naidoo, Universal frames, Topol. Appl., 160 (2013) 2454-2464. [6] Luis Espa˜ nol, Javier Guti´errez Garc´ıa, and Tomasz Kubiak, Separating families of locale maps and localic embeddings, Algebra Universalis 67 (2012) 105-112. [7] V.V. Fedorchuk and A. Chigogidze, Absolyutnye retrakty i beskonechnomernye mnogoobraziya, ”Nauka”, Moscow, 1992 (Russian). [8] S. D. Iliadis, Universal spaces and mappings, North-Holland Mathematics Studies, vol. 198, Elsevier Science B.V., Amsterdam, 2005. [9] Stavros Iliadis, Universal regular and completely regular frames, Topol. Appl., Vol 179(2015) 99-110. [10] S.D. Iliadis, Dimension and Universality on Frames, Topology and its Applications, In print. [11] J.R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972) 5-32. [12] J. Paseka, Boolean algebras and ultracompactness, Cah. Topol. Geom. Differ. Categ. 33 (1992) 15-20. 18
[13] J. Picado and A. Pultr, Frames and locales, Frontiers in Mathematics, Birkh¨ auser/Springer, Basel AG, Basel, 2012. [14] V.G. Vinokurov, A lattice method of defining dimension, Dokl. Akad. Nauk SSSR Tom 168 (1966) 663-666. (Russian) 1
Moscow State Pedagogical University E-mail address: [email protected]
2
Moscow State University (M.V. Lomonosov), Moscow State Pedagogical University E-mail address: [email protected]
3
Moscow State University (M.V. Lomonosov) E-mail address: [email protected]
19
P.S. Gevorgyan, S.D. Iliadis, Yu.V. Sadovnichy
PII: DOI: Reference:
S0166-8641(17)30053-6 http://dx.doi.org/10.1016/j.topol.2017.02.010 TOPOL 6018
To appear in:
Topology and its Applications
Received date: Revised date: Accepted date:
19 January 2016 2 July 2016 20 July 2016
Please cite this article in press as: P.S. Gevorgyan et al., Universality on frames, Topol. Appl. (2017), http://dx.doi.org/10.1016/j.topol.2017.02.010
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Universality on frames P.S. Gevorgyan1 , S.D. Iliadis2 , and Yu.V. Sadovnichy3
Abstract In the present paper we consider so-called saturated classes of frames of weight less than or equal to a given infinite cardinal τ . Such classes are saturated by universal elements. We consider three kinds of saturated classes. The saturated classes of any considered kind have the following basic property: the intersection of not more than τ many saturated classes is also a saturated class and therefore in this intersection there are also universal elements. We prove that the classes RegFrm(τ ) of regular and CRegFrm(τ ) of completely regular frames of weight ≤ τ are saturated classes in any of these kinds. This fact implies that in the classes RegFrm(τ, μ) and CRegFrm(τ, μ) of regular and completely regular frames, respectively, of weight ≤ τ and decomposition invariant (see [10]) μ ≤ τ there are universal elements. Key words: Lattice, Frame, Universal frame, Saturated class of frames, Decomposition invariant of frames. 2000 Mathematics Subject Classification: 06A06, 06D22, 54A05
1. Introduction A topological space T is said to be universal in a class Sp of topological spaces if (a) T ∈ Sp and (b) for each X ∈ Sp there exists a homeomorphism of X into T . If only the second condition is satisfied, then T is called a containing space for Sp. The problem whether there are universal elements actually can be posed for any class of spaces that is determined by a certain topological property. Such problems appeared in topology in its early development, when special classes of separable metrizable spaces were considered. With the consideration of more general classes of spaces, some methods of construction of universal elements appeared. These methods use factorization theorems, separating theorems and product of spaces. In [8] it is described a method of construction of universal and containing spaces in which does not use neither these theorems nor products of spaces. Let Sp be a class of spaces of weight less than or equal to a fixed infinite cardinal τ . The basic element of the method is the notion of a base B for a given collection S of elements of Sp (which is a fixed base of cardinality ≤ τ in each space of the collection) and a family R of equivalence relations on S. Using these two (fixed) elements, a space denoted by T(B, R) is constructed. This space is a containing space for the collection S of spaces. Suppose that for any collection S of elements of Sp some construction containing spaces T(B, R) belong to Sp. Then, T(B, R) will be a universal element in Sp. In the case where the class Sp is saturated by these kind of elements, this class is called saturated. 1
The simplicity of the elements of the above mentioned method of construction of universal elements give us the possibility to use a modification of this method in Lattice Theory and to apply the same scheme for the construction of universal frames. Let Fr be a class of frames. We will say that a frame T is universal in Fr if (a) T ∈ Fr and (b) for each F ∈ F there exists a homomorphism of T onto F . If only the condition (b) is satisfied, then T will be called a containing frame for Fr. Universal elements for different classes of frames (the class of all frames, the class of completely regular frames, the class of regular frames with countable bases, zero-dimensional frames (see [1]) of weight less than or equal to a fixed infinite cardinal are considered (in its localic version) in the paper [6] (see also related papers [11] and [12]). For the construction of universal elements in [5] for the class of all frames of a given infinite weight and in [9] for the classes of all regular frames and of all completely regular frames of any fixed infinite weight it is used a modification of the method of [8]. For the construction of classes of spaces having universal elements the different dimension-like invariants play an important role. Many such classes are defined as classes of spaces for which a given dimension-like invariant takes some concrete value. However, the dimension theory for frames is not sufficiently developed. Although for frames some dimension invariants are defined (see, for example, [2], [3], and [14] ) they do not define classes of frames with universal elements. In the paper [10] the notion of a saturated class of frames is introduced and a dimension-like invariant of a frame, the so-called decomposition invariant, is defined, which is a cardinal (in particular maybe a non-negative integer). It is proved that in the class Frm(τ, μ) of all frames of weight less than or equal to a fixed infinite cardinal τ having decomposition invariant ≤ μ ≤ τ , where μ is a fixed cardinal, is a saturated class and therefore in this class there are universal elements. In the present paper we consider saturated classes of frames of weight less than or equal to a given infinite cardinal τ . Such classes are saturated by universal elements. We consider three kind of saturated classes. The saturated classes of any considered kind have the following basic property: the intersection of not more than τ many saturated classes is also a saturated class and therefore in this intersection there are also universal elements. In Section 3 we prove that the class RegFrm(τ ) of regular and the class CRegFrm(τ ) of completely regular frames of weight ≤ τ are saturated classes in the sense of the paper [10]), which in the present paper are called saturated classes with ∧-mappings. By the basic property of saturated classes and since the above mentioned class Frm(τ, μ) is saturated, this fact implies that the classes RegFrm(τ, μ) and CRegFrm(τ, μ) of regular and completely regular frames, respectively, of weight ≤ τ and decomposition invariant μ ≤ τ , are also saturated classes and therefore in these classes there are universal elements. In Section 4 we introduce the notion of a saturated class of frames by a slightly different way from that of the paper [10]. This new kind of saturated classes, 2
which here are called saturated classes without ∧-mappings, have also the basic property of saturated classes. We prove that the classes RegFrm(τ ) and CRegFrm(τ ) are saturated in this new sense. We prove also that any saturated class without ∧-mappings is a saturated class with ∧-mappings. In Section 5, using the notions of τ -complete sets and its τ -closed subsets (see, for example, [4] or [7]), we introduce another kind of saturated classes having the above mentioned basic property of saturated classes and prove that the classes RegFrm(τ ) and CRegFrm(τ ) are also saturated in this new sense. Although the classes RegFrm(τ ) and CRegFrm(τ ) are saturated in any considered sense, in general it is an open problem if some of these kinds of saturated classes coincide.
2 Preliminaries 2.1 Notation. An ordinal number is considered as the set of smaller ordinal numbers, and a cardinal number is considered as the initial ordinal number of this cardinality. Therefore, for ordinals δ and τ relations δ ∈ τ + 1 and δ ≤ τ are equivalent. By τ we denote a fixed infinite cardinal. The set of all nonempty finite subsets of τ will be denoted by F. The symbol ≡ in a relation means that one or both sides of the relation are new notations. Each mapping f of the set τ onto a set B is called an indication of B and will be denoted by B = {aδ : δ ∈ τ } where aδ = f (δ), δ ∈ τ . Frames. Recall that a frame is a complete lattice L in which x∧ S = {x ∧ s : s ∈ S} for any x ∈ L and any S ⊆ L. Our notations shall be fairly standard from Picado and Pultr [13]. For instance we denote the top element and the bottom element of L by 1L and 0L respectively. For each x ∈ L we put x∗ = {z ∈ L : z ∧ x = 0L }. The correspondence x → x∗ will be called the star operator. For x, y ∈ L we write y ≺ x iff y ∗ ∨x = 1L and y ≺≺ x iff there is a system {cr ∈ L : r ∈ Q∩[0, 1]} (Q is the rational numbers) such that c0 = y, c1 = x and c r ≺ cs whenever r < s. A frame L is said to be regular if for each x ∈ L, x = {y ∈ L : y ≺ x} and completely regular if for each x ∈ L, x = {y ∈ L : y ≺≺ x}. A subset B of a complete lattice (or frame) L is called a base for L if each element of L is a supremum of a subset of B. The weight of a complete lattice (or frame) L is the minimal cardinal κ for which there exists a base B for L of cardinality κ. A frame homomorphism is a map between frames which preserves finite meets, including the top element, and arbitrary joins, including the bottom element.
3
Let L be a class of complete lattices (or frames). We say that an element T ∈ L is universal in this class if for every L ∈ L there exists a homomorphism of T onto L. 2.2 Definitions (see [10]). Let L be an indexed collection of frames (and, therefore, according to our agreements the weights of all elements of L are ≤ τ ). Indexed base for L. Suppose that for every L ∈ L we choose an indexed base of L: B L ≡ {aL δ : δ ∈ τ} L such that aL 0 = 0L and a1 = 1L . Then, the indexed set
B ≡ {B L ≡ {aL δ : δ ∈ τ } : L ∈ L}
(2.1)
will be called a base for L. If all B L , L ∈ L, are closed under taking finite infima and with respect to the star operator, then B is called also closed under taking finite infima and with respect to the star operator. Suppose that for each L ∈ L we have a mapping θ[L] of F into τ such that for L every δ ∈ τ and t ∈ F we have θ[L]({δ}) = δ and aL θ[L](t) = ∧{aδ : δ ∈ t}, then θ[L] will be called the ∧-mapping in L and the set θB ≡ {θ[L] ≡ θB [L] : L ∈ L}
(2.2)
will be called the set of ∧-mappings for B. We note that in general θ[L] is not uniquely determined. Extension of a base. Let B0 ≡ {B 0,L ≡ {a0,L : δ ∈ τ } : L ∈ L}, B1 ≡ {B 1,L ≡ {a1,L : δ ∈ τ } : L ∈ L} δ δ be two bases for L and ϕ a mapping of τ into itself such that ϕ(0) = 0 and ϕ(1) = 1. We say that B1 is an extension of B0 with extension mapping ϕ (from B0 to B1 ) if for every L ∈ L, 0,L a1,L ϕ(δ) = aδ , δ ∈ τ.
In general, the mapping ϕ is not uniquely determined. If the bases B0 and B1 are considered with the sets θB0 ≡ {θB0 [L] : L ∈ L} and θB1 ≡ {θB1 [L] : L ∈ L} of ∧-mappings respectively, then the base B1 is said to be a proper extension of B0 if for every t ∈ F and L ∈ L we have θB1 [L](ϕ(t)) = ϕ(θB0 [L](t)). It is easy to prove that if B1 is a (proper) extension of B0 and a base B2 for L is a (proper) extension of B1 , then B2 is also a (proper) extension of B0 . 4
Moreover, if ϕ0 is a (proper) extension mapping from B0 to B1 and ϕ1 is a (proper) extension mapping from B1 to B2 , then ϕ1 ◦ ϕ0 is a (proper) extension mapping from B0 to B2 . Admissible family of equivalence relations. A family R ≡ {∼s : s ∈ F}
(2.3)
of equivalence relations on L is said to be admissible if: (a) whenever ∅ = p ⊂ s ∈ F we have ∼s ⊂ ∼p and (b) for any s ∈ F the number of equivalence classes of the relation ∼s is less than or equal to τ . We denote by C(∼s ) the set of all equivalence classes of the relation ∼s and put C(R) = ∪{C(∼s ) : s ∈ F}. Note that the cardinality of C(R) is ≤ τ . Final refinement. Let R0 ≡ {∼s0 : s ∈ F} and R1 ≡ {∼s1 : s ∈ F} be two admissible families of equivalence relations on L. It is said that R1 is a final refinement of R0 if for each s ∈ F there exists t ∈ F such that ∼t1 ⊂ ∼s0 . Obviously, if an admissible family R2 is a final refinement of R1 and R1 is a final refinement of R0 , then R2 is a final refinement of the family R0 . The family RB of equivalence relations on L. Let (2.1) be a base for L closed under taking finite infima and considered with the set (2.2) of ∧mappings. We denote by RB ≡ {∼sB : s ∈ F} the family of equivalence relations on L defined as follows: two elements L and L of L are ∼sB -equivalent for some s ∈ F if and only if for each δ, η ∈ s and L L L t ⊂ s we have (a) the relation aL δ ≤ aη implies the relation aδ ≤ aη and (b) θ[L](t) = θ[L ](t). B-admissible family. Let (2.1) be a base for L closed under taking finite infima and considered with the set (2.2) of ∧-mappings. An admissible familly (2.3) of equivalence relations on L is said to be B-admissible if this family is a final refinement of RB . It is easy to see that if R0 is a B-admissible family of equivalence relations on L and R1 is an admissible family, which is a final refinement of R0 , then R1 is also B-admissible. The Containing Frame T(B, R). For any collection L of mutually disjoint frames, an arbitrary base B for L considered with the set of ∧-mappings, and an arbitrary B-admissible family R of equivalence relations on L, the construction of the Containing Frame T(B, R) is given in [10]. In [5] it is given the same construction of T(B, R) considered that R coincides with the family RB . We recall this construction. 5
Let L be a collection of mutually disjoint frames and U the (free) union of elements of L. Further, let (2.1) be a base for L, closed under taking finite infima, and (2.3) a B-admissible family of equivalence relations on L. For every δ ∈ s ∈ F and H ∈ C(∼s ) we put L Aδ (H) = {aL δ : L ∈ H} ∪ {a0 = 0L : L ∈ L \ H} ⊂ U
and
BT = {Aδ (H) : δ ∈ s ∈ F, H ∈ C(∼s )}.
For every L ∈ L and a subset M ≡ {Aδj (Hj ) : j ∈ J} ⊂ BT we put and
L ML = {aL δ ∈ L : {aδ } = Aδ (H) ∩ L, Aδ (H) ∈ M }
= {sup ML : L ∈ L}. M
Now we consider the (non-indexed) set : M ⊆ B}. T ≡ T(B, R) ≡ {M ≤N for M , N ∈ T if and only if for In T we define an order ≤ by putting M every L ∈ L we have sup ML ≤ sup NL . Obviously, the pair (T, ≤) is a poset. ∈ T. The generators of the The subset M ⊂ BT will be called a generator of M elements of T are not uniquely determined. It is easy to see that the union of any number of generators of an element of T is also a generator of this element. Therefore, the union of all generators of a certain element of T is the maximal generator of that element. 2.3 Theorem. (See [10].) The above constructed poset T ≡ T(B, R) is a frame, the set BT is a base of T, and for each L ∈ L the mapping hL : T → L ) = sup ML , M ∈ T, is an onto homomorphism. defined by relation hL (M
3 On regular and completely regular frames Definition of a saturated class of frames. We recall the definition of a saturated class of frames given in [10]. A class Fr of frames is said to be saturated (with respect to the universality) if for any indexed collection L of mutualy disjoint elements of Fr there exists a base B0 (considered with a set of ∧-mappings) for L having the property: for each proper extension B of B0 there exists a Badmissible family RB of equivalence relations on L such that for each admissible family R of equivalence relations on L, which is a final refinement of RB , the Containing Frame T(B, R) (constructed in [10] and presented above) belongs to Fr. The base B0 is called an initial base corresponding to the collection L (and
6
the class Fr) and the family RB is called an initial family corresponding to the base B, the collection L, (and the class Fr). 3.1 Theorem The class RegFrm(τ ) of all regular frames of weight ≤ τ is saturated. Proof. We shall use the notation considered in the proof of Theorem 3.1 of [9]. In particular L is a collection of regular mutually disjoint frames of RegFrm(τ ) such that each element L ∈ RegFrm(τ ) is isomorphic to an element of L and U is the union of elements of L. Also, for each L ∈ L, B L is a regular base of L of cardinality less than or equal to τ closed under taking finite infima and the B L -star operator and (3.1) B L = {aL δ : δ ∈ τ} is a special indication of B L constructed in [9] (see relation (3.1) of [9]). We put B ≡ {{ aL δ : δ ∈ τ } : L ∈ L} and as the set θB = {θL : L ∈ L} of ∧-mappings for B we take the set for which θL = θ for each L ∈ L, where θ is the mapping of F into τ constructed in Theorem 3.1 of [9]. We prove that B is an initial base for L. Indeed, let B be a proper extension of B. We prove that the family RB is an initial family of L corresponding to the base B and the collection L proving simultaneously that B is an initial base for L. It suffices to prove that the Containing Frame T(B , R) is regular, for each admissible family R of equivalence relations on L, which is a final refinement of RB . Let R ≡ {∼s : s ∈ F} be an admissible family of equivalence relations on L, which is a final refinement of RB . By Proposition 4.3 of [10] we have T(B, R) = T(B , R). Therefore, it suffices to prove that the Containing Frame T(B, R) is regular. Recall that the set BT = {Aδ (H) : δ ∈ s ∈ F, H ∈ C(∼s )}, where
Aδ (H) = {aL δ : L ∈ H} ∪ {0L : L ∈ L \ H},
is a base for T ≡ T(B, R). We also consider the Containing Frame T(B, RB ) constructed in [9]. The set {Aδ : δ ∈ τ }, where Aδ = {aL δ : L ∈ L}, is a base for this frame. Here we shall denote this base by B(T). (In [9] this base was denoted by BT .) Since for each δ ∈ τ and any s ∈ F we have Aδ = {Aδ (H) : H ∈ C(∼s )}, each element of B(T) and therefore each element of T(B, RB ) is an element of T. It is clear that the order on T(B, RB ) coincides with the order induced by 7
T. Moreover, the top and the bottom elements of T coincide with the top and bottom elements of T(B, RB ) and will be denoted by 1T and 0T respectively. For each A ∈ B(T) the pseudocomplement of A in T(B, RB ) will be denoted by A∗ (not to be confused with its pseudocomplement A∗ in T). We note that A∗ ≤ A∗ . For each A ≡ {aL A : L ∈ L} and H ∈ C(R) we put A0 (H) = {aL A : L ∈ H} ∪ {0L : L ∈ L \ H}. It is easy to see that A0 (H) is an element of T. Indeed, if A = {Aδ ∈ B(T) : Aδ ∈ M ⊂ B(T)}, then
A0 (H) =
{(Aδ )0 (H) = Aδ (H) ∈ BT : Aδ ∈ M } ∈ T.
To prove the theorem it suffices to prove that for each δ ∈ s ∈ F and any H ∈ C(∼s ) we have (3.2) Aδ (H) = {B ∈ T : Aδ (H) ∨ B ∗ = 1T }. Let δ ∈ s ∈ F and H ∈ C(∼s ). Since the frame T(B, RB ) is regular we have (3.3) Aδ = {A ∈ T(B, RB ) : Aδ ∨ A∗ = 1T }. Obviously, Aδ (H) ∧ Aδ = Aδ (H). Therefore, Aδ (H) = Aδ (H) ∧ Aδ = Aδ (H) ( {A : Aδ ∨ A∗ = 1T } = =
{Aδ (H) ∧ A : Aδ ∨ A∗ = 1T }.
(3.4)
Let A ≡ {aL A : L ∈ L} be an element included in the relation (3.3), Then A ≤ Aδ and therefore (3.5) Aδ (H) ∧ A = A0 (H). Putting
A10 (H) = {0T : L ∈ H} ∪ {1L : L ∈ L \ H}
we have A10 (H) ∈ T and A10 (H) ∧ A0 (H) = 0T . The last relation implies that A10 (H) ≤ (A0 (H))∗ . Therefore, (A0 (H))∗ ∩ L = 1L , L ∈ L \ H.
(3.6)
0
On the other hand: (a) the relation A (H) ≤ A implies that A∗ ≤ A∗ ≤ (A0 (H))∗
(3.7)
L and (b) the relation Aδ ∨ A∗ = 1T implies that aL δ ∨ aA = 1T for each L ∈ L and in particular for each L ∈ H. Therefore, if (A0 (H))∗ = {bL : L ∈ L}, then by relation (3.7) for every L ∈ H, L aL δ ∨ b = 1T .
8
(3.8)
Relations (3.6) and (3.8) imply that Aδ (H) ∨ (A0 (H))∗ = 1T .
(3.9)
Thus, from relations (3.4), (3.5), and (3.9) it follows that Aδ (H) = {A0 (H) : Aδ ∨ A∗ = 1T } ≤ {A0 (H) : Aδ (H) ∨ (A0 (H))∗ = 1T } ≤ ≤
{B ∈ T : Aδ (H) ∨ B ∗ } ≤ Aδ (H)
proving relation (3.2) and completing the prove of the theorem. The proof of the following theorem is similar to the proof of Theorem 3.1. 3.2 Theorem. The class CRegFrm(τ ) of all completely regular frames of weight ≤ τ is saturated. We recall the notion of the decomposition invariant of a frame. Let L be a frame and μ ≤ τ a cardinal. We shall say that the decomposition invariant of L is less than or equal to μ and write dec(L) ≤ μ if there are: (a) zerodimensional frames F η , η ∈ μ + 1 (here, μ is considered as an initial ordinal), and (b) homomorphisms hη of L onto F η , η ∈ μ + 1, such that for each a, b ∈ L the condition hη (a) = hη (b) for every η ∈ μ + 1 implies that a = b. The following corollary follows from the above Theorems 3.1 and 3.2 and Theorems 3.9, 3.10, and 4.7 of [10]. 3.3 Corollary. The class RegFrm(τ, μ) (respectively, the class CRegFrm(τ, μ)) of all regular (respectively, of all completely regular) frames of weight ≤ τ with the decomposition invariant less than or equal to μ ≤ τ is saturated and therefore in this class there are universal elements.
4 On saturated classes Let L be a collection of mutually disjoint frames. In Section 2 for a given base B with a set θB of ∧-mapping we recalled the definition of the family RB of equivalence relations on L and the construction of the Containing Frame T(B, R) for each B-admissible family R of equivalence relations on L. In the present section, in order to distinguish notation, we shall denote such a base B by B[∧] and the family RB by RB[∧] ≡ {∼sB[L] : s ∈ F}. Then, the Containing Frame T(B, R) will be denoted by T(B[∧], R) and therefore the Containing Frame T(B, RB ) by T(B[∧], RB[∧] ). Let
B ≡ {{aL δ : δ ∈ τ } : L ∈ L}
9
(4.1)
be a base for L which is considered without any set of ∧-mappings. We define the family (4.2) RB ≡ {∼sB : s ∈ F} of equivalence relations on L as follows: two elements L and L of L are ∼sB equivalent for some s ∈ F if and only if for each η ∈ s and t ⊂ s the relation L {aL δ : δ ∈ t} = aη implies the relation
L {aL δ : δ ∈ t} = aη .
L L L Therefore, for δ ∈ s the relation aL δ ≤ aη implies the relation aδ ≤ aη .
As in Section 2 an admissible family R ≡ {∼s : s ∈ F}
(4.3)
of equivalence relations on L is said to be B-admissible if for each s ∈ F there exists t ∈ F such that ∼t ⊂ ∼sB . Let (4.1) be a base for L and (4.3) a B-admissible family of equivalence relations on L. Then, we can define a poset T(B, R) exactly as in Section 2 we define the Containing Frame T(B, R). (We note that in that definition we did not use the set of ∧-mappings for B. This set is used for the definition of the family RB , which determines what families R we can consider). Although Theorem 4.1 given below has the same formulation of Theorem 2.3 it does not follow from this Theorem because the families R in both theorems are distinct in general. 4.1 Theorem. The poset T ≡ T(B, R) is a frame, the set BT is a base of T, and ) = sup ML , for each L ∈ L the mapping hL : T → L defined by relation hL (M M ∈ T, is an onto homomorphism. Proof. The proof of the theorem is similar to that of Theorem 1.3 of [5]. It follows from the following three lemmas. Lemma 1. The poset (T, ≤) is a complete lattice of weight ≤ τ . Lemma 2. For every L ∈ L there exists a homomorphism hL of T onto L. Lemma 3. The complete lattice (T, ≤) is a frame. The proofs of Lemmas 1 and 3 are the same as the proofs of the corresponding lemmas of Theorem 1.3 of [5]. Although the proof of Lemma 2 follows the proof
10
of Lemma 2 of Theorem 1.3 we shall present here a part of this proof since some notions, which have the same notations, are different. The proof of Lemma 2. Let L be a fixed element of L. The definition of the mapping hL and the proofs of the facts that it is onto and preserves suprema are the same as in Lemma 2 of the above mentioned theorem. We shall prove that the mapping hL preserves finite infima. Let A1 ≡ Aδj (Hj ) and A2 ≡ Aηi (Hi ) be two fixed elements of BT such that δj ∈ sj , Hj ∈ R(∼sj ), ηi ∈ si , and Hi ∈ R(∼si ). We shall define a subset of BT , denoted by M (A1 , A2 ) ≡ Aδj (Hj )ΔAηi (Hi ) as follows. If the set Hj ∩ Hi is empty, then we put M (A1 , A2 ) = {0T }. If Hj ∩ Hi = ∅, then we put M (A1 , A2 ) = {Aδ (H) : {δ} ∪ sj ∪ si ⊂ s ∈ F, H ∈ C(∼s ),
L L H ⊂ Hj ∩ Hi , aL δj ∧ aδi = aδ for each L ∈ H}
It is clear that for every Aδ (H) ∈ M (A1 , A2 ) we have Aδ (H) ≤ Aδj (Hj ) and Aδ (H) ≤ Aηi (Hi ). Moreover, it is easy to verify that if δ ≡ δj = δi and Hj = Hi , then Aδj (Hj ) = Aηi (Hi ) ≤ {Aδ (H ) ∈ M (A1 , A2 ) : H ∈ C(∼sj ∪si )}.
(4.4)
(4.5)
and K be two elements of T. We need to prove that Now, let N ∧ K) = sup NL ∧ sup KL . hL (N
(4.6)
Suppose that N = {Aδj (Hj ) : j ∈ J, Hj ∈ C(∼sj ), δj ∈ sj ∈ F} and
K = {Aηi (Hi ) : i ∈ I, Hi ∈ C(∼si ), ηi ∈ si ∈ F}.
Without loss of generality we can suppose that N and K are the maximal and K, respectively. generators of N Consider the set M≡
{Aδj (Hj )ΔAηi (Hi ) : (j, i) ∈ J × I}
11
. First, we shall prove that and denote by Mmax the maximal generator of M =N ∧ K. M
(4.7)
Obviously, for every (j, i) ∈ J × I we have and Aη (Hi ) ≤ K. Aδj (Hj ) ≤ N i
(4.8)
Then, since N and K are maximal generators, by the definition of the set M and the relations (4.4) and (4.8), for every (j, i) ∈ J × I we have Aδj (Hj )ΔAηi (Hi ) ⊂ N ∩ K and, therefore, M ⊂ N and M ⊂ K. This means that sup ML ≤ sup NL and sup ML ≤ sup KL .
(4.9)
Relation (4.9) implies that ≤N and M ≤ K. M Now suppose that for an element P ∈ T we have and P ≤ K. P ≤ N Since N and K are maximal generators we have P ⊂ N and P ⊂ K. Let Aδ (H) be an element of P . Then, Aδ (H) ∈ N ∩K. This means that for some j ∈ J we have δ = δj , H = Hj ∈ C(∼sj ), and therefore A1 ≡ Aδ (H) = Aδj (Hj ). Similarly, for some i ∈ I we have δ = ηi , H = Hi ∈ C(∼si ), and therefore A2 ≡ Aδ (H) = Aηi (Hi ). In this case, all Aδ (H ) included in formula (4.5), as elements of M (A1 , A2 ), are elements of M . This means that Aδ (H) = Aδj (Hj ) . Thus, P ⊂ Mmax and therefore is an element of the maximal generator of M completing the proof of the relation (4.7). P ≤ M We shall now prove that sup ML = sup NL ∧ sup KL .
(4.10)
(Recall that L is a fixed element of L.) First we shall prove that sup NL ∧ sup KL ≤ sup ML .
(4.11)
Let b ∈ NL , c ∈ KL , and a = b ∧ c. It suffices to prove that a ∈ ML . This is clear if a = 0L . Suppose that a = 0L . There exist A1 ≡ Aδj (Hj ) ∈ N with Hj ∈ C(∼sj ) and A2 ≡ Aηi (Hi ) ∈ K with Hi ∈ C(∼si ) such that b = aL δj , c = L L aηi , and L ∈ Hj ∩ Hi . There exists δ ∈ τ such that a = aδ . Let s = {δ} ∪ sj ∪ si 12
and H be the element of C(∼s ) containing L. Then, Aδ (H) ∈ M (A1 , A2 ) ⊆ M which means that a = aL δ ∈ ML proving relation (4.11). Now we prove that sup ML ≤ sup NL ∧ sup KL .
(4.12)
Let a ∈ ML . It suffices to find b ∈ NL and c ∈ KL such that a ≤ b ∧ c. For a = 0L we can take b = c = 0L . Therefore we can suppose that a = 0L . There exist δj ∈ sj ∈ F, Hj ∈ C(∼sj ), ηi ∈ si ∈ F and Hi ∈ C(∼si ) such that for some δ ∈ τ and H ∈ C(∼s ), where s = {δ} ∪ sj ∪ si , we have L ∈ H, a = aL δ , and Aδ (H) ∈ Aδj (Hj )ΔAηi (Hi ). L L L The last relation means that aL δ = aδj ∧ aηi . Therefore, we can put b = aδj and c = aL ηi completing the proof of relation (4.12) and therefore relation (4.10). Relations (4.7) and (4.10) imply relation (4.6) completing the proofs of the lemma and theorem.
The following proposition follows immediately from the construction of Containing Frames T(B, R) and T(B[∧], R). 4.2 Proposition. Suppose that the base B is considered with a set of ∧mappings and the family (4.3) is B-admissible and B[∧]-admissible. Then, the Containing Frames T(B, R) and T(B[∧], R) coincide. Saturated classes without ∧-mappings. Now we give a notion of a saturated class using bases for a collection of frames without the sets of ∧-mappings. A class Fr of frames is said to be saturated if for every collection L of elements of Fr there exists a base B0 for L satisfying the following property: for each extension B of B0 there exists a B-admissible family RB of equivalence relations on L such that for each admissible family R of equivalence relations on L, which is a final refinement of RB , the Containing Frame T(B, R) belongs to Fr. In this case the base B0 is called an initial base corresponding to the collection L of elements of Fr and the family RB will be called an initial family corresponding to the base B and the collection L. Saturated classes in the above sense will be called saturated classes without ∧-mappings and saturated classes considered in Section 2 will be called saturated classes with ∧-mappings. 4.3 Theorem. Any saturated class of frames without ∧-mappings is a saturated class with ∧-mappings. Proof. Let Fr be a saturated class of frames without ∧-mappings. Consider an arbitrary collection L of elements of Fr and let B0 be an initial base corresponding to the collection L. Now, consider an arbitrary set of ∧-mappings for the base B0 . We shall prove that the base B0 [∧], that is, the base B0 considered 13
with the set of ∧-mappings, is an initial base (with respect to saturated classes with ∧-mapings) for the collection L. Indeed, let B[∧] be a proper extension of B0 [∧]. Then, the base B is an extension of B0 . Therefore, there exists a B-admissible family RB which is initial corresponding to the base B and the collection L. Denote by RB[∧] an admissible family of equivalence relations on L, which is simultaneously a final refinement of the families RB and RB[∧] . We prove that RB[∧] is initial corresponding to the base B[∧] and the collection L proving simultaneously that B[∧] is an initial base. Indeed, let R be an admissible family of equivalence relations on L, which is a final refinement of RB[∧] . Then, R is a final refinement of RB , which means that the Containing Frame T(B, R) belongs to Fr. By Proposition 4.2, T(B, R) = T(B[∧], R) and therefore T(B[∧], R) ∈ Fr proving that RB[∧] is initial corresponding to the base B[∧] and completing the proof of the theorem. The proof of the next theorem is similar to that of Theorem 3.1. We note that in Theorem 3.1 the set of ∧-mappings is use only for the construction of the initial base for L. 4.4 Theorem. The classes of regular and completely regular frames of weight ≤ τ are saturated classes without ∧-mappings. The proof of the following theorem is the same as the one of Theorem 3.10 of [10] 4.5 Theorem. The intersection of not more than τ many saturated classes without ∧-mappings is also a saturated class without ∧-mappings.
5. On a new type of saturated classes of frames τ -Complete sets (see, for example, [4] or [7]). Let C be a directed set (that is, for each c1 , c2 ∈ C there exists c ∈ C such that c1 ≤ c and c2 ≤ c). For a subset K ⊂ C we denote by sup(K) an element c ∈ C (if such an element exists) with the properties: (a) k ≤ c for each k ∈ K and (b) if c is an element of C such that k ≤ c for each k ∈ K, then c ≤ c. Obviously, if there exists sup(K), then it is uniquely determined. The set C is said to be τ -complete if for each chain K ⊂ C (that is, K is a linearly ordered subset) of cardinality ≤ τ there exists sup(K) ∈ C. A subset D of C is said to be τ -closed if for each chain K ⊂ D of cardinality ≤ τ we have sup(K) ∈ D. A subset K is said to be cofinal if for each e ∈ C there exists k ∈ K such that e ≤ k. We recall the following lemma. 5.1 Lemma. The intersection of not more than τ many non-empty τ -closed cofinal subsets of a τ -complete set C is also a non-empty τ -closed cofinal subset of C. 14
Agreement. In this section we suppose that L is a collection of mutually disjoint frames of weight ≤ τ . It is also assumed that any considered base B for L has cardinality ≤ τ and closed under taking finite infima. All bases for L will be considered without the set of ∧-mappings. The sets P ≡ P(L) and P ≡ P(L). We denote by P ≡ P(L) the set of all pairs (B, R), where B is a base for L and R is a B-admissible family of equivalence relations on L. On this set we a define a preorder ≤ setting that (B1 , R1 ) ≤ (B2 , R2 ) iff B2 is an extension of B1 and R2 is a final refinement of R1 . This preorder defines on L the natural equivalence relation, denoted by ∼, that is (B1 , R1 ) ∼ (B2 , R2 ) iff (B1 , R1 ) ≤ (B2 , R2 ) and (B2 , R2 ) ≤ (B1 , R1 ). The equivalence classes of the relations ∼ will be denoted by P ≡ P(L). The equivalence class containing an element (B, R) ∈ P will be denoted by [(B, R)]. On the set P we define an order ≤ setting [(B1 , R1 )] ≤ [(B2 , R2 )] iff (B1 , R1 ) ≤ (B2 , R2 ). It is easy to prove that the order ≤ on P is well defined, and that is it is independent of the pairs (B1 , R1 ) and (B2 , R2 ) of elements [(B1 , R1 )] and [(B2 , R2 )] respectively. 5.3 Proposition. The set P is τ -complete. Proof. It is easy to see that P is directed. Let K be a subset of P of cardinality ≤ τ (in particular, K may be a chain). We must prove that there is sup(K) ∈ P. In particular, this fact will imply that P is directed. Consider an arbitrary indication of K: K ≡ {[(Bδ , Rδ )] : δ ∈ τ }, where and
Bδ ≡ {{aL δ,ε : ε ∈ τ } : L ∈ L} Rδ ≡ {∼sδ : s ∈ F}.
Let χ : τ × τ → τ be a bijection. We define a base B ≡ {{aL η : η ∈ τ } : L ∈ L} L −1 (η). Therefore, the mapping ϕδ : for L setting aL η = aδ,ε , where (δ, ε) = χ τ → τ for which ϕδ (ε) = χ(δ, ε) , ε ∈ τ , is an extension mapping from Bδ to B. Also, we define an admissible family
R ≡ {∼s : s ∈ F} of equivalence relations on L by setting ∼s = ∩{∼sδ : δ ∈ s}. We prove that the pair (B, R) is an element of P and [(B, R)] = sup(K).
15
First we prove that (B, R) ∈ P, that is R is B-admissible. Let s ∈ F. We must prove that there exists t ∈ F such that ∼t ⊂ ∼sB . Let s ≡ {η1 , ..., ηn } ∈ F, i = 1, ..., n, and
χ−1 (ηi ) = (δi , εi ), i = 1, ..., n.
Since P is directed there exists δ ∈ τ such that [(Bδi , Rδi )] ≤ [(Bδ , Rδ )], i = 1, ..., n. Let ϕi be an extension mapping from Bδi to Bδ and ϕi (δi , εi ) = (δ , εi ), i = i, ..., n. We put Then,
s ≡ {εi : i = 1, ..., n} ∈ F. L L aL ηi = aδi ,εi = aδ ,εi , i = 1, ..., n, L ∈ L.
The last relation implies that ∼sB = ∼sBδ . Since R is a final refinement of Rδ
and Rδ is a final refinement of RBδ there exists t ∈ F such that ∼t ⊂ ∼sBδ and therefore ∼t ⊂ ∼sB proving that R is a final refinement of RB .
Now, we prove that [(B, R)] = sup(K). Let (B∗ , R∗ ), where R∗ = {∼s∗ : s ∈ F}, be an element of P such that [(Bδ , Rδ )] ≤ [(B∗ , R∗ )], δ ∈ τ, that is B∗ is an extension of Bδ with an extension mapping ϕδ∗ and R∗ is a final refinement of Rδ . We must prove that [(B, R)] ≤ [(B∗ , R∗ )], that is B∗ is an extension of B and for each s ∈ F there exists t ∈ F such that ∼t∗ ⊂ ∼s . It is easy to see that the mapping ϕ∗ : τ → τ such that for each η ∈ τ , ϕ∗ (η) = ϕδ∗ (ε), where (δ, ε) = χ−1 (η), is an extension mapping from B to B∗ , that is B∗ is an extension of B. Let s ∈ F. By the definition of the equivalence relation ∼s we have ∼s = ∩{∼sδ : δ ∈ s}. For each δ ∈ s there exists tδ ∈ τ such that ∼t∗δ ⊂ ∼sδ . Since R∗ is admissible there exists t ∈ F such that ∼t∗ ⊂ ∼t∗δ for each δ ∈ s and therefore ∼t∗ ⊂ ∩{∼t∗δ : δ ∈ s} ⊂ ∼s completing the proof of the proposition.
16
5.4 Proposition. Let [(B1 , R1 )] = [(B2 , R2 )] ∈ P. Then, the Containing Frames T1 ≡ T(B1 , R1 ) and T2 ≡ T(B2 , R2 ) coincide. Proof. Since B2 is an extension of B1 by Lemma 4.3 of [10] it follows that T(B1 , R2 ) = T(B2 , R2 ). (We note that in Lemma 4.3 of [10] we actually did not use the set of ∧-mappings.) Therefore it suffices to prove that T(B1 , R1 ) = T(B1 , R2 ).
(5.1)
Let s s B1 ≡ {{aL δ : δ ∈ τ } : L ∈ L}, R1 ≡ {∼1 : s ∈ F}, and R2 ≡ {∼2 : s ∈ F}.
Consider the bases BT1 ≡ {Aδ (H) : δ ∈ s ∈ F, H ∈ C(∼s1 )} and
BT2 ≡ {Bδ (G) : δ ∈ s ∈ F, G ∈ C(∼s2 )}
of T1 and T2 , respectively, where Aδ (H) = {aL δ : L ∈ H} ∪ {0L : L ∈ L \ H} and
Bδ (G) = {aL δ : L ∈ G} ∪ {0L : L ∈ L \ G}.
Let Aδ (H) be an arbitrary fixed element of BT1 , where H ∈ C(∼s1 ). Since R2 is a final refinement of R1 there exists t ∈ F such that ∼t2 ⊂ ∼s1 . Therefore, H = ∪{Gi : i ∈ I} ⊂ C(∼t2 ). Then, Aδ (H) = {Aδ (Gi ) : i ∈ I} proving that T1 ⊂ T2 . Similarly, T2 ⊂ T1 . Therefore, T1 = T2 which completes the proof of the proposition. Thus, if e ≡ [(B, R)] ∈ P, then we can use the notation T(e) ≡ T(B, R). Saturated class of frames by τ -complete sets. A class Fr of frames is said to be saturated (by τ -complete sets) if for every indexed collection L of mutually disjoint elements of Fr the set of all elements e ∈ P(L) for which T(e) ∈ Fr contains a τ -closed cofinal subset of P(L). 5.5 Theorem. The classes of regular and completely regular frames of weight ≤ τ are saturated by τ -complete sets. Proof. Consider the class RegFrm(τ ) and let L ba a collection of elements of RegFrm(τ ). In Theorem 3.1 we proved that there exists a base B for L such that for each extension B of B the Containing Space T(B , RB ) is an element 17
of RegFrm(τ ). This means that setting e ≡ [(B , RB )] and e = [(B, RB )] we have e ≤ e and T(e ) ∈ RegFrm(τ ). Then, the proof of the theorem follows from the fact that the subset {e ∈ P(L) : e ≤ e } is τ -closed. The following theorem follows from Lemma 5.1. 5.6 Theorem. The intersection of not than τ many saturated classes of frames by τ -complete sets is also a saturated class by τ -complete sets and therefore in this intersection there are universal elements. 5.7 Problem. Do some of the considered kinds of saturated classes coincide?
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[13] J. Picado and A. Pultr, Frames and locales, Frontiers in Mathematics, Birkh¨ auser/Springer, Basel AG, Basel, 2012. [14] V.G. Vinokurov, A lattice method of defining dimension, Dokl. Akad. Nauk SSSR Tom 168 (1966) 663-666. (Russian) 1
Moscow State Pedagogical University E-mail address: [email protected]
2
Moscow State University (M.V. Lomonosov), Moscow State Pedagogical University E-mail address: [email protected]
3
Moscow State University (M.V. Lomonosov) E-mail address: [email protected]
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