On Some Generalizations of the Concept of Partition

There are well-known isomorphisms between the complete lattice of all partitions of a given set A and the lattice of all equivalence relations on A. In the paper the notion of partition is generalized in order to work correctly for wider classes of binary relations than equivalence ones such as quasiorders or tolerance relations. Some others classes of binary relations and corresponding counterparts of partitions are considered.


Introduction
A partition of a given set A is any family Π of subsets of A such that: (i) any member of the family is nonempty and (ii) each element a of the set A belongs to exactly one set X from Π. Given the partitions Π 1 , Π 2 of A we say that Π 1 is a refinement of (or is finer than) Π 2 iff ∀X ∈ Π 1 ∃Y ∈ Π 2 , X ⊆ Y . The relation of refinement, denoted as ≤, turns out to be a partial order on the set P art ( We are going to apply two methods to obtain new classes of families of subsets of a given set A (other than the one of all partitions of A). These new classes when equipped with refinement-like relations, form posets, which we shall show are isomorphic to the posets composed of appropriate classes of binary relations defined on A, ordered by inclusion. In special cases, when a class Θ ⊆ P (A × A) (P is the operation of powerset) of binary relations on A and a class F ⊆ P (P (A)) of families of subsets of A are such that E(A) ⊆ Θ and P art(A) ⊆ F, the isomorphism f : (Θ, ⊆) −→ (F, ≤) should fulfil the expected condition that the restriction f to E(A) is just the above mentioned isomorphism f . It turns out, as a result of the first method, that several classes of binary relations defined on a set A, such as quasiorders, so-called antiquasiorders and semiequivalence relations, can be associated with appropriate classes of families of subsets of A. The second method leads to two results: firstly, we obtain two different isomorphisms joining the class of so-called semitolerance relations defined on A with two classes of families of subsets of A; and secondly, we provide a similar result for tolerance relations. This result is important since the tolerance relations are used to form the "quotient" algebras, in particular lattices composed of the "blocks" of "partitions", leading to the important method of gluing of lattices, cf. for example [6,9]. Such quotient modulo tolerances algebras are considered without any justification, i.e. without providing a form of isomorphism joining the class of all tolerances with an appropriate class of families of "blocks", cf. [1,[3][4][5]11]. According to our knowledge, [2] is the only paper which gives an appropriate family of blocks, called the τ -covering of a set, corresponding to a given tolerance relation. However, the conditions defining the notion of τ -covering of a set are rather non-intuitive and complicated, in comparison with our more natural approach to describe a "partition" corresponding to a tolerance.
Although some of the results presented here can be generalized also to the case A = ∅, it is assumed in the whole paper (sometimes explicitly) that the set A on which the considered relations are defined and whose subsets form "partitions" of different types, is nonempty.

Preliminaries Concerning the First Method
The following simple observation is basic for the first method of searching for other types of families of subsets of a given set A which are isomorphic to classes of binary relations defined on A. The isomorphism f : E(A) −→ P art(A), is a composition of two mappings: the first, φ : E(A) −→ P (A) A , assigns to each equivalence relation ρ the canonical mapping k ρ : A −→ P (A) defined by k ρ (a) = [a] ρ . The second assigns to each k ∈ P (A) A its counterdomain k[A]. The first mapping, φ, can be generalized to be defined on the set of all binary relations on A, as follows. First, for any ρ ⊆ A × A and any a ∈ A, we define the equivalential class of a to be the set (a] ρ = {x ∈ A : xρa}. Now let φ(ρ) be the function defined on A by φ(ρ)(a) = (a] ρ . This defines a function φ : P (A×A) −→ P (A) A . We note that (P (A×A), ⊆) is a complete lattice, under the usual set inclusion, and that (P (A) A , ≤) is also a complete lattice, under the ordering given by k 1 ≤ k 2 iff for each a ∈ A we have k 1 (a) ⊆ k 2 (a).
From now on we will consider the correspondence A × A ⊇ ρ =⇒ k ρ ∈ P (A) A , where for each a ∈ A, k ρ (a) = (a] ρ , and the inverse correspondence: Moreover, with each map k : A −→ P (A) we can associate its counterdo- In general, the assignment: ∈ P (P (A)) is obviously neither 1-1 nor onto. However, for some special subsets of P (A) A and of P (P (A)) the correspondence is in fact 1-1 and onto. In such a case, when there is a 1-1 and onto correspondence between a set K ⊆ P (A) A and P ⊆ P (P (A)) one can establish immediately that the posets ({ρ k : k ∈ K}, ⊆), (P, ≤ ) are isomorphic. Here, by definition, for any R 1 , R 2 ∈ P : R 1 ≤ R 2 iff k 1 ≤ k 2 , where k 1 corresponds to R 1 , and k 2 to R 2 . The isomorphism of these posets is a composition of two isomorphisms: , where φ over the arrow denotes the restriction of the isomorphism φ from Proposition 1 to the set ψ[K] of binary relations on A. Obviously, the final form of the isomorphism is as follows: ρ k −→ {(a] ρ k : a ∈ A}, for each k ∈ K. For example, consider the set P art(A) in a role of the subset P of P (P (A)) and a subset K of P (A) A composed of all the mappings k fulfilling the following conditions: (1) ∀a ∈ A, a ∈ k(a), or, equivalently, the following one: Then given k ∈ K, the family k[A] is a partition of A. Conversely, given any Π ∈ P art(A) define the mapping k Π : A −→ P (A) by setting k Π (a) to be the single element of Π to which a belongs, for each a ∈ A. Then the mapping k Π fulfils the conditions (1) and (2). Moreover, for each k ∈ K we have k k[A] = k and for any Π ∈ P art(A) we have k Π [A] = Π. Thence the correspondence: K k =⇒ k[A] ∈ P art(A), is 1-1 and onto. In this way we see that the posets ({ρ k : k ∈ K}, ⊆), (P art(A), ≤ ) where for any , i = 1, 2 are isomorphic. However, one can show that {ρ k : k ∈ K} = E(A) (to this aim it is convenient to use the condition (3) rather than (1) and (2)); moreover, for each k 1 , k 2 ∈ K : . So finally, using this method, we rediscover the well-known theorem we started from. Now we are going to apply the method to find the analogous connections between some other classes of binary relations defined on a given set A and the corresponding classes of families of subsets of A.

Quasiorders and Quasipartitions
Definition. A family R of subsets of a given set A will be said to be a quasipartition of the set A iff In other words, a family R ⊆ P (A) is a quasipartition of A iff R = { R a : a ∈ A}. (The name "quasi-partition" already occurs in the literature: in [5] it is used in an informal way as the name of the partition counterpart corresponding to a tolerance relation, which in [2] is referred to as the τ -covering of a set.) As a simple example of a quasipartition, consider a nonempty set A and Notice that any ordinary partition Π of a set A is a quasipartition of A.
Here for each a ∈ A, Π a is a singleton. Now consider the class of all the mappings k : A −→ P (A) fulfilling the following conditions: It is easily seen that given such a function k from that class, its counterdomain k[A] is a quasipartition of A (the condition (k 2) means: ∀a ∈ A, k(a) = k[A] a ). Conversely, given any quasipartition R, define a function k R : A −→ P (A) by setting k R (a) = R a , for each a ∈ A. Then the function k R fulfils the conditions (k 1), (k 2). The condition (k 1) holds due to the fact that for each a ∈ A we have a ∈ R a , i.e., a ∈ k R (a). The inclusion (⊇) of (k 2) follows due to the same fact. The inclusion (⊆) is equivalent to the expression: ∀a, Furthermore, for any mapping k such that the conditions (k 1), (k 2) are satisfied we have k k[A] = k. On the other hand, for any quasipartition R of A we have k R [A] = R. This means that there exists a 1-1 correspondence between the class of all the quasipartitions and the class of all mappings k fulfilling (k 1), (k 2). Now it is sufficient to read the conditions (k 1), (k 2) in terms of corresponding binary relations in order to obtain an appropriate connection between the quasipartitions and a suitable class of relations. So the condition (k 2) is equivalent to the following one: which holds for a relation ρ if and only if ρ is a quasiorder on A (that is a reflexive and transitive binary relation on A). The condition (k 1) is equivalent to the condition: ∀a ∈ A∃x ∈ A aρx (which means by definition that ρ is serial), therefore yields nothing more to (qo).
Let QOrd(A), QPart(A) are the sets of all quasiorders and of all quasipartitions of a set A, respectively. Then the following connection follows: Obviously we have just stated that the poset (QP art(A), ≤) is a complete lattice as it is isomorphic to the complete lattice (QOrd(A), ⊆) in which A × A and the identity relation id A on A are the greatest and the least elements respectively, and for any nonempty Θ ⊆ QOrd(A) we have inf Θ = Θ and supΘ = Θ, where for any binary relation ρ defined on A the relation ρ is the transitive closure of ρ.
One may show that the isomorphism from Corollary 2 restricted to the class E(A) of all the equivalence relations on A is just the isomorphism

Antiquasiorders and Dual Quasipartitions
Definition. A family R of subsets of a given set A will be said to be a dual quasipartition of the set A iff The word "dual" is used here in the following sense: Proof.
(⇒) : Assume that R is a dual quasipartition of A. We show that S = {−X : X ∈ R} is a quasipartition of A. To this aim consider any X ∈ R.
Therefore, any element from S is of the form S a for some a ∈ A. On the other hand, for any a ∈ A, Thus any element of R is of the form: R a for some a ∈ A. On the other hand, for any Now consider the class of all the mappings k : A −→ P (A) fulfilling the following condition: It is easily seen that given a function k satisfying condition (k ), its counterdomain k[A] is a dual quasipartition of A. Conversely, a function k R : A −→ P (A) defined by a dual quasipartition R in the following way: ∀a ∈ A, k R (a) = R a , fulfils the condition (k ). Furthermore, for any mapping k such that (k ) is satisfied, k k[A] = k. On the other hand, for any dual quasipartition R of A we have k R [A] = R. This means that there exists a 1-1 correspondence between the class of all the dual quasipartitions and the class of all mappings k fulfilling (k ). Now, one can simply rewrite the condition (k ) in terms of binary relations ρ corresponding to the maps k which satisfy (k ), in the following way: for any a, b ∈ A, bρa iff ∃x ∈ A (¬aρx & bρx). This condition is equivalent to the following one: which in turn is equivalent to the conjunction of two conditions: Let us call such a binary relation ρ on A fulfilling (i), (ii) an antiquasiorder, which is justified by the following: Proof. Take any quasiorder ρ defined on A. Then the following condition (equivalent to the statement that ρ is a quasiorder) holds: ). This is equivalent to the following: ∀a, b ∈ A (bρa iff ∀x ∈ A (¬bρx ⇒ ¬aρx)), which means that the complement −ρ (= A 2 − ρ) fulfils the condition (aqo). Conversely, consider any antiquasiorder ρ on A. Then from (aqo) it follows that for any a, b One can show that the class of all binary relations ρ fulfilling the condition (ii) is just the class of all complements of transitive relations defined on A. So let us call a relation ρ from that class intransitive.
Notice that a binary relation ρ defined on A may be both transitive and intransitive: Notice also that the poset (AQOrd(A), ⊆) of all antiquasiorders defined on a set A is a complete lattice such that (A × A) − id A and the empty relation are the greatest and the least elements, respectively, and for any Given any antiquasiorder −ρ, where ρ is a quasiorder on A (cf. Fact 4) for any a ∈ A we have (a] −ρ = −(a] ρ . Therefore, the canonical mapping k −ρ corresponding to the relation −ρ is of the form: for each a ∈ A, k −ρ (a) = −(a] ρ . In this way the dual quasipartition corresponding to the antiquasiorder −ρ is of the form: {k −ρ (a) : a ∈ A} = {−(a] ρ : a ∈ A} (cf. Fact 3). In turn, given any dual quasipartition R of A let R = {−X : X ∈ R} be the corresponding quasipartition of A. Moreover, let −ρ 1 be the antiquasiorder corresponding to R, that is R = {(a] −ρ 1 : a ∈ A}. Then obviously, R = {(a] ρ 1 : a ∈ A}. Suppose similarly that a dual quasipartition S = {(a] −ρ 2 : a ∈ A}, where ρ 2 is a quasiorder. Then for the relation of refinement of dual quasipartitions the following holds: Finally, one can also consider all the complements of equivalence relations on A. They form the class of all irreflexive, intransitive and symmetric relations on A hereafter called antiequivalence relations. Furthermore, the class Restricting the isomorphism from Corollary 6 to the class AE(A) one obtains the following:

Semiequivalence Relations and Pseudopartitions
Definition. A family R of subsets of a given set A will be said to be a pseudopartition of the set A iff where, as previously, for each a ∈ A, R a = {X ∈ R : a ∈ X}. In other words, a family This definition leads to the following characterization of a pseudopartition of a set A: Proof. Let R be any pseudopartition of a set A and a ∈ A. Suppose that X ∈ R a .
(⊇) : Since a ∈ X, so a ∈ R a and, finally, R a ∈ R a .
It is evident that given any element of A, in the family of all members of a given pseudopartition of A containing that element, either there is exactly one set or there is none. So each element of A either belongs to exactly one set in the pseudopartition or does not belong to any of its sets. The difference between partition and pseudopartition of a set A consists only in the fact that the empty set may be an element of a pseudopartition while it cannot be a member of a partition. Now consider the class of all the mappings k : A −→ P (A) fulfilling the following condition: It is evident that given a function k from that class, its counterdomain k[A] is a pseudopartition of A. Conversely, a function k R : A −→ P (A) defined by a pseudopartition R by ∀a ∈ A, k R (a) = R a , fulfils the condition (−k ) because for any Furthermore, for any mapping k such that the condition (−k ) is satisfied, for any a ∈ A, we have On the other hand, for any pseudopartition This means that there exists a 1-1 correspondence between the class of all the pseudopartitions and the class of all mappings k fulfilling (−k ). Now, the condition (−k ) can be expressed in terms of binary relations as follows: Let us call a binary relation ρ defined on A semiequivalence iff ρ is symmetric, transitive and semireflexive, where the semireflexive property is: Obviously, a semiequivalence relation defined on a set A, restricted to the set {a ∈ A : aρa} (of all elements of A which are not isolated with respect to ρ) is an ordinary equivalence relation on that set. Furthermore, we have the following Fact 9. For any binary relation ρ on A, ρ is a semiequivalence iff the condition (se) holds for ρ.

Proof.
(⇒) : Suppose that ρ is a semiequivalence on A. Then the condition (se)(⇒) follows due to semireflexivity of ρ. The converse condition: (se)(⇐) holds due to symmetry and transitivity of ρ.
(⇐) : Assume (se). Then the symmetry of ρ follows directly. Next we have that for each b ∈ A : bρb iff ∃x ∈ A, bρx, which together with symmetry implies semireflexivity of ρ. For transitivity, suppose that bρa and aρc. Then we have cρa from symmetry. In this way, ∃x ∈ A (bρx & cρx). So bρc follows from (se).

One can check that the class SE(A) of all the semiequivalence relations defined on A forms a complete lattice. Its greatest element is A × A,
and its least element is ∅; for any nonempty family Θ ⊆ SE(A), we have inf Θ = Θ and supΘ = Θ. Thus (E(A), ⊆) forms a complete sublattice of (SE(A), ⊆).
Notice also that applying Fact 8, given two functions k 1 , k 2 fulfilling the condition (−k ) one can easily show that where R, S are the pseudopartitions of A which are counterdomains of k 1 and k 2 , respectively.
Taking into account the considerations of the section one can formulate the following result.

Preliminaries Concerning the Second Method
The second method used to establish the counterparts of partitions related with tolerance relations is based on two tools. The first one is a general correspondence between the family of all downward subsets with maximal elements in a given poset and the family of all antichains of the poset: Lemma 11. Given any poset (A, ≤), let U ⊆ A fulfil the following conditions:

where for any B ⊆ A, M ax(B) is the set of all maximal elements in (B, ≤).
Proof. Straightforward. Now it is sufficient to show that for any U 1 , U 2 ∈ DW M (A) we have (⇒): When U 1 ⊆ U 2 and x ∈ Max(U 1 ) then from (c) applied to U 2 it follows that there exists a ∈ Max(U 2 ) such that x ≤ a.
(⇐): Suppose that ∀x ∈ Max(U 1 )∃y ∈ Max(U 2 ), x ≤ y, and let x ∈ U 1 . Then x ≤ a for some a ∈ Max(U 1 ) by (c). Hence and from the assumption it follows that x ≤ b for some b ∈ Max(U 2 ). So x ∈ U 2 by Lemma 11.
The second tool of the method is the concept of a residuated pair of mappings. We are going to apply it first, in order to provide one of the counterparts of partition corresponding to the so-called semitolerance relation. Let us recall briefly the notion. Given any posets (A, ≤ A ), (B, ≤ B ), a pair of mappings f : or equivalently, f and g are monotone and for any a ∈ A, b ∈ B: a and b ≤ B f (g(b)).
Some authors use the adjective "residuated" in case the converse orderings instead of ≤ A , ≤ B are applied. A function f : A −→ B for which there exists a map g : B −→ A such that (f, g) is residuated pair, is also called residuated. Given a residuated function there is a unique g such that (f, g) is a residuated pair. Given a residuated pair (f, g) of functions, their compositions I and C of the form: Ia = g(f (a)), Cb = g(f (b)), any a ∈ A, b ∈ B, are interior and closure operations on A and B, respectively. The following condition is also satisfied: In this way it follows that the mapping f restricted to the image g [B] is a bijection from that image onto f [A]. Moreover, for any a 1 , a 2 ∈ g[B] : (cf. for example [7,8]).  defined on A, for  each a, b ∈ A, (a, b)

Semitolerance Relations and Corresponding "Partitions"
. The closure operation C : P (P (A)) −→ P (P (A)) is of the form: for any Let us call a semitolerance any binary relation on A which is semireflexive and symmetric. The name takes its origin from the tolerance relations introduced in [10] as reflexive and symmetric relations, and considered in many papers.
Notice that for any binary relation ρ on A, the relation I(ρ) is a semitolerance. Moreover, given any semitolerance ρ on A, one may easily show that ρ ⊆ I(ρ). In this way the following lemma holds: Lemma 15. For any binary relation ρ defined on A, ρ is an open element, that is I(ρ) = ρ, iff ρ is a semitolerance relation.
In order to characterize the closed elements in the lattice (P (P (A)), ⊆), first let us formulate the following obvious fact.
Fact 16. Given any S ⊆ P (A) and X ⊆ A the following conditions are equivalent: Lemma 17. For any S ⊆ P (A), S is closed, that is C(S) = S, iff the following two conditions hold: Proof. Notice that according to the definition of the closure operation C, the condition that S is closed is equivalent to the following one: for each (⇒): Assume that S is closed. In order to show (a) let ∀Z ∈ P 2 (X), Z ∈ S, where X ⊆ A. Then ∀Z ∈ P 2 (X)∃Y ∈ S, Z ⊆ Y . So X 2 ⊆ {Y 2 : Y ∈ S} due to Fact 16 and consequently, X ∈ S by the assumption. In order to show (b) suppose that X ⊆ Y and Y ∈ S. Then obviously X 2 ⊆ Y 2 , therefore X 2 ⊆ {U 2 : U ∈ S}, thus X ∈ S due to the assumption.
(⇐): Assume that (a) and (b) hold and suppose that X 2 ⊆ {Y 2 : Y ∈ S}. Having (a), in order to complete the proof, it is enough to show that P 2 (X) ⊆ S. So let Z ∈ P 2 (X). Then from Fact 16 and the third assumption it follows that for some Y ∈ S, Z ⊆ Y , which due to (b) leads to the result Z ∈ S.
Corollary 18. Let A be any set. Then  ⊆). In this way, a candidate to be a counterpart of partition corresponding to a given semitolerance ρ defined on A is just the family f (ρ) = {X ⊆ A : X 2 ⊆ ρ}. In order to see the difference between an ordinary partition and its counterpart, let us consider the latter when it corresponds to an equivalence relation: Proof. Let θ be an equivalence relation on A.
(⊇) : Suppose that X ⊆ [a] θ for some a ∈ A. Let x, y ∈ X. Then x, y ∈ [a] θ , so (x, y) ∈ θ. This means that X 2 ⊆ θ, that is X ∈ f (θ). Evidently, the ordinary partition related with an equivalence relation θ is the family of all the maximal elements of f (θ). We are going to generalize this result for all the semitolerances, introducing the second counterpart of partition which coincides with the ordinary partition in case of the equivalence relations. As is well known (cf. e.g. [5,9]), in case of a tolerance relation θ defined on a given lattice, such maximal elements of the family f (θ) play the role of the blocks of a "partition" (they are elements of a quotient lattice modulo a tolerance).   (P (A)), ). Taking into account Corollary 18(i), it is easily seen that the composition of that new isomorphism and the isomorphism f from the corollary, is the isomorphism of the complete lattice (ST (A), ⊆) of all the semitolerance relations on A onto that partially ordered subset. So the antichains from the subset form just the counterparts (of the second kind) of partitions corresponding to semitolerances. We will focus our attention on showing that this subset is composed of all the antichains B of (P (A), ⊆) which fulfil the following condition: It coincides with the condition (2) of the definition of τ -covering of a set in [2] whenever the expression "two-element subset" used in [2] also allows singleton sets to be included (cf. the next section 8).
Therefore from the assumption we obtain that P 2 (Z) ⊆ S. However, the condition (a) from Lemma 17 is satisfied for S. Thus Z ∈ S which implies that Z ⊆ X for some X ∈ Max(S).  (P (A)), ) and G from (Ach * (P (A)), ) onto (ST (A), ⊆), respectively. Simply, taking into account Proposition 19 one can see that in case ρ is an equivalence relation on A we have (cf. the proof of the corollary) Conversely, when Π is an ordinary partition of A then obviously, Π ∈ Ach * (P (A)), and G(Π) = {X 2 : X ∈ Π} which is the equivalence relation induced by the partition Π.
In order to establish a connection between a "partition" F (ρ), given a semitolerance ρ on A, and an equivalential class (a] ρ = {x ∈ A : xρa} let us extend the mapping F to the whole class P (A × A) of binary relations on A : F (ρ) = Max({X ⊆ A : X 2 ⊆ ρ}, and formulate some simple facts:

Proof.
For (1): Suppose that X = {(a] ρ : a ∈ X}. Then directly: X 2 ⊆ ρ. In order to show that X is a maximal set with the property: X 2 ⊆ ρ, assume, conversely, that this does not hold. Then there is a Y ⊆ A such that X ⊆ Y, X = Y and Y 2 ⊆ ρ. So b ∈ X for some b ∈ Y . Thus according to the assumption there exists an a ∈ X such that bρa does not hold. However, a ∈ Y so bρa is true; a contradiction.
For (2): When X ⊆ A is such that X 2 ⊆ ρ, then obviously, X ⊆ {(a] ρ : a ∈ X}. For (3): Assume that ρ is a nonempty semitolerance relation. Due to (1), (2) it suffices to show the implication: X ∈ F (ρ) implies that {(a] ρ : a ∈ X} ⊆ X. So suppose that X ∈ F (ρ) and conversely that there is an element b ∈ {(a] ρ : a ∈ X} such that b ∈ X. Then for each a ∈ X, bρa, and from the symmetry of ρ : for each a ∈ X, aρb. Moreover, since ρ = ∅ so ∅ ∈ F (ρ) which together with assumption imply that X = ∅. Therefore bρb, due to the condition (sr) of semireflexivity. From the very assumption it follows that X 2 ⊆ ρ. All this means that (X ∪ {b}) 2 ⊆ ρ and consequently, X ∈ F (ρ).
Obviously, Σ(A) ⊆ Σ(A). One may show that the value g(S) of the isomorphism g from Corollary 18(i) on the family S ∈ Σ(A) is just a tolerance relation. Conversely, given any ρ ∈ T (A), where T (A) is the class of all the tolerances defined on A, we have f (ρ) ∈ Σ(A), where f is the isomorphism from that corollary. So remembering that T (A) forms a complete lattice (T (A), ⊆) such that for each nonempty Θ ⊆ T (A), supΘ = Θ and inf Θ = Θ, sup∅ = id A , inf∅ = A 2 , one obtains the following Similarly, one obtains a counterpart of Corollary 23 for the tolerance relations. Namely, consider a subset Ach * (P (A)) of Ach * (P (A)) consisting of all the antichains B included in (P (A), ⊆) which not only fulfil the condition (*) but also the condition: In [2] a 1-1 and onto correspondence between T (A) and the class of all so-called τ -coverings of A, was established. In our notation, it is just the correspondence G from Corollary 26: a tolerance ρ corresponds to a τ -covering B of A iff ρ = G(B). Indeed, it is evident that the set Ach * (P (A)) and the family of all τ -coverings of A, coincide. In our notation, a τ -covering of a nonempty set A is a family B of subsets of A such that the conditions (*) and (d) are satisfied, as well as the condition (Instead of (*) the following clause occurs in [2]: (2) if N ⊆ A and N is not contained in any set from B, then N contains a two-element subset of the same property.