The Monoid Consisting of Kuratowski Operations

Abstract

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphism types are applied. Some results apply only to topology, and one cannot bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.

1. Introduction

Let 𝑋 be a topological space. Denote by 𝐴⁻ closure of the set 𝐴⊆𝑋. Let 𝐴^𝑐 be the complement of 𝐴; that is, 𝑋⧵𝐴=𝐴^𝑐. The aim of this paper is to examine monoids generated under compositions from the closure and the complement. A widely known fact due to Kuratowski [1] states that at most 14 distinct operations can be formed from such compositions. Mark them as follows. Kuratowski operations: 𝜎₀(𝐴)=𝐴 (the identity), 𝜎₁(𝐴)=𝐴^𝑐 (the complement), 𝜎₂(𝐴)=𝐴⁻  (the closure),  𝜎₃(𝐴)=𝐴^𝑐−,  𝜎₄(𝐴)=𝐴^−𝑐,  𝜎₅(𝐴)=𝐴^𝑐−𝑐(the interior),  𝜎₆(𝐴)=𝐴^−𝑐−,  𝜎₇(𝐴)=𝐴^{𝑐−𝑐−},  𝜎₈(𝐴)=𝐴^{−𝑐−𝑐},  𝜎₉(𝐴)=𝐴^{𝑐−𝑐−𝑐},  𝜎₁₀(𝐴)=𝐴^{−𝑐−𝑐−},  𝜎₁₁(𝐴)=𝐴^{𝑐−𝑐−𝑐−},  𝜎₁₂(𝐴)=𝐴^{−𝑐−𝑐−𝑐},  𝜎₁₃(𝐴)=𝐴^{𝑐−𝑐−𝑐−𝑐}. The following rules was found in the original paper by Kuratowski [1, pages 183-184]. In the Engelking book [2], they are commented by a hint on page 81. Cancellation rules: 𝐴^−𝑐−=𝐴^{−𝑐−𝑐−𝑐−},𝐴^{𝑐−𝑐−}=𝐴^{𝑐−𝑐−𝑐−𝑐−}. (1)

Kuratowski operations have been studied by several authors, for example, [3] or [4]. A list of some other authors one can find in the paper [5] by Gardner and Jackson. For the first time these operations were systematically studied in the dissertation by Kuratowski, whose results were published in [1]. Tasks relating to these operations are usually resolved at lectures or exercises with general topology. They are normally left to students for independent resolution. For example, determine how many different ways they convert a given set.

This note is organized as follows. Kuratowski operations and their marking are described in the introduction. Their properties of a much broader context than for topologies are presented in Section 2. The Cayley table, for the monoid 𝕄 of all Kuratowski operations, has been drawn up in Section 3. We hope that this table has not yet been published in the literature. Having this table, one can create a computer program that calculates all the semigroups contained in 𝕄. However, in Sections 4–8, we present a framework (i.e., techniques using only paper and pencil) to point out all 118 semigroups and 56 isomorphism types of them. The list of 43 semigroups which are not monoids is presented in Section 9. In this section, also isomorphism types are discussed in order of the number of elements in semigroups. Finally, we present cancellation rules (relations) motivated by some topological spaces.

2. Cancellation Rules

A map 𝑓∶𝑃(𝑋)→𝑃(𝑋) is called: (i)increasing, if 𝐴⊆𝐵 implies 𝑓(𝐴)⊆𝑓(𝐵); (ii)decreasing, if 𝐴⊆𝐵 implies 𝑓(𝐵)⊆𝑓(𝐴); (iii)an involution, if the composition 𝑓∘𝑓 is the identity; (iv)an idempotent, if 𝑓∘𝑓=𝑓. Assume that 𝐴↦𝜎₀(𝐴) is the identity, 𝐴↦𝜎₁(𝐴) is a decreasing involution, and 𝐴↦𝜎₂(𝐴) is an increasing idempotent map. Other operations 𝜎_𝑖 let be compositions of 𝜎₁ and 𝜎₂ as it was with the kuratowski operations. We get the following cancellation rules.

Lemma 1. If 𝐵⊆𝜎₂(𝐵), then 𝜎₂∘𝜎₁₂=𝜎₆,𝜎₂∘𝜎₁₃=𝜎₇. (2)

Proof. For clarity of this proof, use designations 𝜎₁(𝐴)=𝐴^𝑐 and 𝜎₂(𝐴)=𝐴⁻. Thus, we shall prove 𝐴^{−𝑐−𝑐−𝑐−}=𝐴^−𝑐−,𝐴^{𝑐−𝑐−𝑐−𝑐−}=𝐴^{𝑐−𝑐−}. (3) We start with 𝐴^{−𝑐−𝑐}⊆𝐴^{−𝑐−𝑐−}, substituting 𝐵=𝐴^{−𝑐−𝑐} in 𝐵⊆𝐵⁻. This corresponds to 𝐴^{−𝑐−𝑐−𝑐}⊆𝐴^{−𝑐−𝑐𝑐}=𝐴^−𝑐−, since 𝜎₁ is a decreasing involution. Hence 𝐴^{−𝑐−𝑐−𝑐−}⊆𝐴^−𝑐−, since 𝜎₂ an increasing idempotent.
Since 𝜎₁ is decreasing, 𝜎₂ is increasing, and 𝐵⊆𝐵⁻, we have 𝐵^−𝑐⊆𝐵^𝑐⊆𝐵^𝑐−. (4) Thus 𝐴^{−𝑐−𝑐}⊆𝐴^{−𝑐𝑐−}=𝐴⁻, if we put 𝐴^−𝑐=𝐵. Again using that 𝜎₂ is increasing and 𝜎₁ is decreasing, we obtain 𝐴^{−𝑐−𝑐−}⊆𝐴⁻ and then 𝐴^−𝑐⊆𝐴^{−𝑐−𝑐−𝑐}. Finally, we get 𝐴^−𝑐−⊆𝐴^{−𝑐−𝑐−𝑐−}.
With 𝐴^𝑐 in the place of 𝐴 in the rule 𝐴^−𝑐−=𝐴^{−𝑐−𝑐−𝑐−} we get the second rule.

In academic textbooks of general topology, for example, [2, Problem 1.7.1.], one can find a hint suggested to prove the above cancellation rules. Students go like this: steps 𝐴⊆𝐴⁻ and 𝐴^𝑐⊆𝐴^𝑐− lead to 𝐴^𝑐−𝑐⊆𝐴; the special case 𝐴^{−𝑐−𝑐−𝑐}⊆𝐴^−𝑐− (of 𝐴^𝑐−𝑐⊆𝐴) leads to 𝐴^{−𝑐−𝑐−𝑐−}⊆𝐴^−𝑐−; steps 𝐴^{−𝑐−𝑐}⊆𝐴⁻ and 𝐴^{−𝑐−𝑐−}⊆𝐴⁻ lead to 𝐴^−𝑐−⊆𝐴^{−𝑐−𝑐−𝑐−}; at the end, use the last step of the proof of Lemma 1. Note that, the above proofs do not use the axioms of topology as follows: (i)∅=∅⁻; (ii)(𝐴∪𝐵)⁻=𝐴⁻∪𝐵⁻. In the literature there are articles in which Kuratowski operations are replaced by some other mappings. For example, Koenen [6] considered linear spaces and put 𝜎₂(𝐴) to be the convex hull of 𝐴. In fact, Shum [7] considered 𝜎₂ as the closure due to the algebraic operations. Add to this, that these operations can be applied to so-called Fréchet (V) spaces, which were considered in the book [8, pages 3–37].

3. The Monoid 𝕄

Let 𝕄 be the monoid consisting of all Kuratowski operations; that is, there are assumed cancellation rules: 𝜎₆=𝜎₂∘𝜎₁₂ and 𝜎₇=𝜎₂∘𝜎₁₃. Fill in the Cayley table for 𝕄, where the row and column marked by the identity are omitted. Similarly as in [9], the factor that labels the row comes first, and that the factor that labels the column is second. For example, 𝜎_𝑖∘𝜎_𝑘 is in the row marked by 𝜎_𝑖 and the column marked by 𝜎_𝑘𝜎₁𝜎₂𝜎₃𝜎₄𝜎₅𝜎₆𝜎₇𝜎₈𝜎₉𝜎₁₀𝜎₁₁𝜎₁₂𝜎₁₃𝜎₁𝜎₀𝜎₄𝜎₅𝜎₂𝜎₃𝜎₈𝜎₉𝜎₆𝜎₇𝜎₁₂𝜎₁₃𝜎₁₀𝜎₁₁𝜎₂𝜎₃𝜎₂𝜎₃𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₃𝜎₂𝜎₆𝜎₇𝜎₂𝜎₃𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₄𝜎₅𝜎₄𝜎₅𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₅𝜎₄𝜎₈𝜎₉𝜎₄𝜎₅𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₇𝜎₆𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₉𝜎₈𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₁₀𝜎₁₁𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₁₁𝜎₁₀𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₁₀𝜎₁₁𝜎₆𝜎₇𝜎₆𝜎₇𝜎₁₀𝜎₁₁𝜎₁₂𝜎₁₃𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₁₃𝜎₁₂𝜎₈𝜎₉𝜎₁₂𝜎₁₃𝜎₁₂𝜎₁₃𝜎₈𝜎₉𝜎₈𝜎₉𝜎₁₂𝜎₁₃ (5)

It turns out that the above table allows us to describe all semigroups contained in 𝕄, using pencil-and-paper techniques, only. The argument will be by a systematic study of possibilities. Preparing the list of all semigroups consisting of Kuratowski operations we used following principles: (i) minimal collection of generators is written using ⟨𝐴,𝐵,…,𝑍⟩, where letters denote generators; (ii) when a semigroup has a few minimal collections of generators, then its name is the first collection in the dictionary order; (iii) all minimal collections of generators are written with the exception of some containing 𝜎₀;(iv) we leave to the readers verification that our list is complete, sometimes we add hints.

4. Semigroups with 𝝈_𝟏

Observe that each semigroup which contains 𝜎₀ is a monoid. Since 𝜎₁∘𝜎₁=𝜎₀, a semigroup which contains 𝜎₁ is a monoid, too.

Theorem 2. There are three monoids containing 𝜎₁: (1)⟨𝜎₁⟩={𝜎₀,𝜎₁}; (2)𝕄=⟨𝜎₁,𝜎_𝑖⟩={𝜎₀,𝜎₁,…,𝜎₁₃}, where 𝑖∈{2,3,4,5}; (3) let 𝕄₁=⟨𝜎₁,𝜎₆⟩. If 𝑗∈{6,7,…,13}, then 𝕄₁=𝜎₁,𝜎_𝑗=𝜎₀,𝜎₁∪𝜎₆,𝜎₇,…,𝜎₁₃. (6)

Proof. The equality ⟨𝜎₁⟩={𝜎₀,𝜎₁} is obvious.
Since 𝜎₂=𝜎₃∘𝜎₁=𝜎₁∘𝜎₄=𝜎₁∘𝜎₅∘𝜎₁, we have ⟨𝜎₁,𝜎₂⟩=⟨𝜎₁,𝜎₃⟩=⟨𝜎₁,𝜎₄⟩=⟨𝜎₁,𝜎₅⟩=𝕄.
If 𝑗∈{6,7,…,13}, then any composition 𝜎_𝑗∘𝜎_𝑖 or 𝜎_𝑘∘𝜎_𝑗 belongs to 𝕄₁, and so ⟨𝜎₁,𝜎_𝑗⟩⊆𝕄₁. We have 𝜎₇=𝜎₆∘𝜎₁, 𝜎₈=𝜎₁∘𝜎₆, 𝜎₉=𝜎₁∘𝜎₇, 𝜎₁₀=𝜎₆∘𝜎₆, 𝜎₁₁=𝜎₆∘𝜎₇, 𝜎₁₂=𝜎₈∘𝜎₆, and 𝜎₁₃=𝜎₈∘𝜎₇, and so 𝕄₁=⟨𝜎₁,𝜎₆⟩={𝜎₀,𝜎₁}∪{𝜎₆,𝜎₇,…,𝜎₁₃}.
Since 𝜎₆=𝜎₇∘𝜎₁=𝜎₁∘𝜎₈=𝜎₁∘𝜎₉∘𝜎₁=𝜎₁₀∘𝜎₁∘𝜎₁₀=𝜎₁₁∘𝜎₁₁∘𝜎₁=𝜎₁∘𝜎₁₂∘𝜎₁₂ and 𝜎₆=𝜎₁∘𝜎₁₃∘𝜎₁∘𝜎₁₃∘𝜎₁, we have ⟨𝜎₁,𝜎_𝑗⟩=𝕄₁ for 𝑗∈{6,7,…,13}.

Consider the permutation 𝜎₀𝜎₁𝜎₂𝜎₃𝜎₄𝜎₅𝜎₆𝜎₇𝜎₈𝜎₉𝜎₁₀𝜎₁₁𝜎₁₂𝜎₁₃𝜎₀𝜎₁𝜎₅𝜎₄𝜎₃𝜎₂𝜎₉𝜎₈𝜎₇𝜎₆𝜎₁₃𝜎₁₂𝜎₁₁𝜎₁₀. (7) It determines an automorphism 𝔸∶𝕄→𝕄.

Theorem 3. The identity and 𝔸 are the only automorphisms of 𝕄.

Proof. Delete rows and columns marked by 𝜎₁ in the Cayley table for 𝕄. Then, check that the operation 𝜎₃ is in the row or the column marked by 𝜎₃ only. Also, the operation 𝜎₄ is in the row or the column marked by 𝜎₄ only. Therefore the semigroup 𝜎₃,𝜎₄=𝜎₂,𝜎₃,…,𝜎₁₃ (8) has a unique minimal set of generators {𝜎₃,𝜎₄}. The reader is left to check this with the Cayley table for 𝕄.
Suppose 𝔾 is an automorphism of 𝕄. By Theorem 2, 𝔾 transforms the set {𝜎₂,𝜎₃,𝜎₄,𝜎₅} onto itself. However 𝜎₂ and 𝜎₅ are idempotents, but 𝜎₃ and 𝜎₄ are not idempotents. So, there are two possibilities: 𝔾(𝜎₃)=𝜎₃ and 𝔾(𝜎₄)=𝜎₄, which implies that 𝔾 is the identity; 𝔾(𝜎₃)=𝜎₄ and 𝔾(𝜎₄)=𝜎₃, which implies 𝔾=𝔸. The reader is left to check this with the Cayley table for 𝕄. We offer hints: 𝜎₂=𝜎₃∘𝜎₄, 𝜎₅=𝜎₄∘𝜎₃, 𝜎₆=𝜎₃∘𝜎₂, 𝜎₇=𝜎₃∘𝜎₃, 𝜎₈=𝜎₄∘𝜎₄, 𝜎₉=𝜎₄∘𝜎₅, 𝜎₁₀=𝜎₆∘𝜎₆, 𝜎₁₁=𝜎₆∘𝜎₇, 𝜎₁₂=𝜎₈∘𝜎₆, and 𝜎₁₃=𝜎₈∘𝜎₇, to verify the details of this proof.

5. The Monoid of All Idempotents

The set {𝜎₀,𝜎₂,𝜎₅,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃} consists of all squares in 𝕄. These squares are idempotents and lie on the main diagonal in the Cayley table for 𝕄. They constitute a monoid and 𝜎₀,𝜎₂,𝜎₅,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃=𝜎₀,𝜎₂,𝜎₅. (9) The permutation 𝜎₀𝜎₂𝜎₅𝜎₇𝜎₈𝜎₁₀𝜎₁₃𝜎₀𝜎₂𝜎₅𝜎₈𝜎₇𝜎₁₀𝜎₁₃ (10) determines the bijection 𝕀∶⟨𝜎₀,𝜎₂,𝜎₅⟩→⟨𝜎₀,𝜎₂,𝜎₅⟩, such that 𝕀𝛼∘𝛽=𝕀𝛽∘𝕀(𝛼), (11) for any 𝛼,𝛽∈⟨𝜎₀,𝜎₂,𝜎₅⟩. To verify this, apply equalities 𝜎₂∘𝜎₅=𝜎₇, 𝜎₅∘𝜎₂=𝜎₈, 𝜎₂∘𝜎₅∘𝜎₂=𝜎₁₀, and 𝜎₅∘𝜎₂∘𝜎₅=𝜎₁₃. Any bijection 𝕀∶𝐺→𝐻, having property 𝕀(𝛼∘𝛽)=𝕀(𝛽)∘𝕀(𝛼), transposes Cayley tables for semigroups 𝐺 and 𝐻. Several semigroups contained in 𝕄 have this property. We leave the reader to verify this.

We shall classify all semigroups contained in the semigroup ⟨𝜎₂,𝜎₅⟩. Every such semigroup can be extended to a monoid by attaching 𝜎₀ to it. This gives a complete classification of all semigroups in ⟨𝜎₀,𝜎₂,𝜎₅⟩.

The semigroup ⟨𝜎₂,𝜎₅⟩ contains six groups with exactly one element.

Semigroups ⟨𝜎₂,𝜎₁₀⟩={𝜎₂,𝜎₁₀} and ⟨𝜎₅,𝜎₁₃⟩={𝜎₅,𝜎₁₃} are monoids. Both consist of exactly two elements and are not groups, so they are isomorphic.

Semigroups ⟨𝜎₇,𝜎₁₀⟩ and ⟨𝜎₈,𝜎₁₃⟩ are isomorphic, in particular 𝔸[⟨𝜎₇,𝜎₁₀⟩]=⟨𝜎₈,𝜎₁₃⟩. Also, semigroups ⟨𝜎₇,𝜎₁₃⟩ and ⟨𝜎₈,𝜎₁₀⟩ are isomorphic by 𝔸. Every of these four semigroups has exactly two elements. None of them is a monoid. They form two types of nonisomorphic semigroups, because the bijections {(𝜎₇,𝜎₁₀),(𝜎₁₀,𝜎₈)} and {(𝜎₇,𝜎₈),(𝜎₁₀,𝜎₁₀)} are not isomorphisms.

Semigroups ⟨𝜎₂,𝜎₇⟩={𝜎₂,𝜎₇,𝜎₁₀} and ⟨𝜎₅,𝜎₈⟩={𝜎₅,𝜎₈,𝜎₁₃} are isomorphic. Also, semigroups ⟨𝜎₂,𝜎₈⟩={𝜎₂,𝜎₈,𝜎₁₀} and ⟨𝜎₅,𝜎₇⟩={𝜎₅,𝜎₇,𝜎₁₃} are isomorphic. In fact, 𝔸[⟨𝜎₂,𝜎₇⟩]=⟨𝜎₅,𝜎₈⟩ and 𝔸[⟨𝜎₂,𝜎₈⟩]=⟨𝜎₅,𝜎₇⟩. None of these semigroups is a monoid. They form two types of nonisomorphic semigroups. Indeed, any isomorphism between ⟨𝜎₂,𝜎₇⟩ and ⟨𝜎₂,𝜎₈⟩ must be the identity on the monoid ⟨𝜎₂,𝜎₁₀⟩. Therefore would have to be the restriction of 𝕀. But 𝕀 restricted to ⟨𝜎₇,𝜎₁₀⟩ is not an isomorphism.

Semigroups ⟨𝜎₂,𝜎₁₃⟩=⟨𝜎₂,𝜎₇,𝜎₈⟩={𝜎₂,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃} and ⟨𝜎₅,𝜎₁₀⟩=⟨𝜎₅,𝜎₇,𝜎₈⟩={𝜎₅,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃} are not monoids. They are isomorphic by 𝔸.

The semigroup ⟨𝜎₂,𝜎₅⟩ contains exactly one semigroup with four elements ⟨𝜎₇,𝜎₈⟩=⟨𝜎₁₀,𝜎₁₃⟩={𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃} which is not a monoid.

Note that ⟨𝜎₂,𝜎₅⟩ contains twenty different semigroups with nine non-isomorphism types. These are six isomorphic groups with exactly one element ⟨𝜎₂⟩≅⟨𝜎₅⟩≅⟨𝜎₇⟩≅⟨𝜎₈⟩≅⟨𝜎₁₀⟩≅⟨𝜎₁₃⟩, two isomorphic monoids with exactly two elements ⟨𝜎₂,𝜎₁₀⟩≅⟨𝜎₅,𝜎₁₃⟩, two pairs of isomorphic semigroups with exactly two elements ⟨𝜎₇,𝜎₁₀⟩≅⟨𝜎₈,𝜎₁₃⟩ and ⟨𝜎₇,𝜎₁₃⟩≅⟨𝜎₈,𝜎₁₀⟩, two pairs of isomorphic semigroups with exactly three elements ⟨𝜎₂,𝜎₇⟩≅⟨𝜎₅,𝜎₈⟩ and ⟨𝜎₂,𝜎₈⟩≅⟨𝜎₅,𝜎₇⟩, a semigroup with exactly four elements ⟨𝜎₇,𝜎₈⟩, and two isomorphic semigroups with exactly five elements ⟨𝜎₂,𝜎₁₃⟩≅⟨𝜎₅,𝜎₁₀⟩ and also ⟨𝜎₂,𝜎₅⟩. Thus, ⟨𝜎₂,𝜎₅⟩ contains twenty different semigroups with nine isomorphism types. But ⟨𝜎₀,𝜎₂,𝜎₅⟩ contains forty-one different semigroups with seventeen isomorphism types. Indeed, adding 𝜎₀ to semigroups contained in ⟨𝜎₂,𝜎₅⟩, which are not monoids, we get twenty monoids with eight non-isomorphism types. Adding 𝜎₀ to a group contained in ⟨𝜎₂,𝜎₅⟩ we get a monoid isomorphic to ⟨𝜎₂,𝜎₁₀⟩.

6. The Semigroup Consisting of {𝝈_𝟔,𝝈_𝟕,…,𝝈_𝟏𝟑}

Using the Cayley table for 𝕄, check that 𝔸𝜎₆,𝜎₇,…,𝜎₁₃=𝜎₆,𝜎₇,…,𝜎₁₃. (12) Similarly, check that the semigroup ⟨𝜎₆,𝜎₉⟩={𝜎₆,𝜎₇,…,𝜎₁₃} can be represented as ⟨𝜎₆,𝜎₁₃⟩, ⟨𝜎₇,𝜎₁₂⟩, ⟨𝜎₈,𝜎₁₁⟩, ⟨𝜎₉,𝜎₁₀⟩, or ⟨𝜎₁₁,𝜎₁₂⟩. Also ⟨𝜎₆,𝜎₉⟩=⟨𝜎₆,𝜎₇,𝜎₈⟩=⟨𝜎₇,𝜎₈,𝜎₉⟩=⟨𝜎₁₀,𝜎₁₁,𝜎₁₃⟩=⟨𝜎₁₀,𝜎₁₂,𝜎₁₃⟩. These representations exhaust all minimal collections of the Kuratowski operations which generate ⟨𝜎₆,𝜎₉⟩. Other semigroups included in ⟨𝜎₆,𝜎₉⟩ have one or two minimal collection of generators. One generator has groups ⟨𝜎₆⟩={𝜎₆,𝜎₁₀}, ⟨𝜎₉⟩={𝜎₉,𝜎₁₃}, ⟨𝜎₁₁⟩={𝜎₇,𝜎₁₁}, and ⟨𝜎₁₂⟩={𝜎₈,𝜎₁₂}. Each of them has exactly two elements, so they are isomorphic. Semigroups ⟨𝜎₇,𝜎₁₀⟩, ⟨𝜎₇,𝜎₁₃⟩, ⟨𝜎₈,𝜎₁₀⟩, ⟨𝜎₈,𝜎₁₃⟩, and ⟨𝜎₇,𝜎₈⟩ are discussed in the previous section. Contained in ⟨𝜎₆,𝜎₉⟩ and not previously discussed semigroups are ⟨𝜎₆,𝜎₇⟩, ⟨𝜎₆,𝜎₈⟩,⟨𝜎₇,𝜎₉⟩, and ⟨𝜎₈,𝜎₉⟩. We leave the reader to verify that the following are all possible pairs of Kuratowski operations which constitute a minimal collection of generators for semigroups contained in ⟨𝜎₆,𝜎₉⟩, but different from the whole. One has (i)⟨𝜎₆,𝜎₇⟩=⟨𝜎₆,𝜎₁₁⟩=⟨𝜎₁₀,𝜎₁₁⟩={𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}; (ii)⟨𝜎₆,𝜎₈⟩=⟨𝜎₆,𝜎₁₂⟩=⟨𝜎₁₀,𝜎₁₂⟩={𝜎₆,𝜎₈,𝜎₁₀,𝜎₁₂}; (iii)⟨𝜎₇,𝜎₉⟩=⟨𝜎₉,𝜎₁₁⟩=⟨𝜎₁₁,𝜎₁₃⟩={𝜎₇,𝜎₉,𝜎₁₁,𝜎₁₃}; (iv)⟨𝜎₈,𝜎₉⟩=⟨𝜎₉,𝜎₁₂⟩=⟨𝜎₁₂,𝜎₁₃⟩={𝜎₈,𝜎₉,𝜎₁₂,𝜎₁₃}.

Proposition 4. Semigroups ⟨𝜎₆,𝜎₇⟩ and ⟨𝜎₈,𝜎₉⟩ are isomorphic, and also semigroups ⟨𝜎₆,𝜎₈⟩ and ⟨𝜎₇,𝜎₉⟩ are isomorphic, but semigroups ⟨𝜎₆,𝜎₇⟩, ⟨𝜎₆,𝜎₈⟩ are not isomorphic.

Proof. Isomorphisms are defined by 𝔸. Suppose 𝐽∶⟨𝜎₆,𝜎₇⟩→⟨𝜎₆,𝜎₈⟩ is an isomorphism. Thus 𝐽[⟨𝜎₇,𝜎₁₀⟩]=⟨𝜎₈,𝜎₁₀⟩. Given 𝐽(𝜎₇)=𝜎₈, we get 𝜎₁₀=𝐽(𝜎₁₀)=𝐽(𝜎₇∘𝜎₁₀)≠𝜎₈∘𝜎₁₀=𝜎₈. But 𝐽(𝜎₇)=𝜎₁₀ implies 𝜎₁₀=𝐽(𝜎₇)=𝐽(𝜎₁₀∘𝜎₇)≠𝜎₈∘𝜎₁₀=𝜎₈. Both possibilities lead to a contradiction.

So, ⟨𝜎₆,𝜎₉⟩ contains eighteen different semigroups with eight non-isomorphism types. Indeed, these are four isomorphic groups with exactly one element ⟨𝜎₇⟩≅⟨𝜎₈⟩≅⟨𝜎₁₀⟩≅⟨𝜎₁₃⟩, four isomorphic groups with exactly two elements ⟨𝜎₆⟩≅⟨𝜎₉⟩≅⟨𝜎₁₁⟩≅⟨𝜎₁₂⟩, two pairs of isomorphic semigroups with exactly two elements ⟨𝜎₇,𝜎₁₀⟩≅⟨𝜎₈,𝜎₁₃⟩ and ⟨𝜎₇,𝜎₁₃⟩≅⟨𝜎₈,𝜎₁₀⟩, and five semigroups with exactly four elements ⟨𝜎₆,𝜎₇⟩≅⟨𝜎₈,𝜎₉⟩, ⟨𝜎₆,𝜎₈⟩≅⟨𝜎₇,𝜎₉⟩,  ⟨𝜎₇,𝜎₈⟩, and also ⟨𝜎₆,𝜎₉⟩.

7. Remaining Semigroups in ⟨𝝈_𝟑,𝝈_𝟒⟩

We have yet to discuss semigroups included in ⟨𝜎₃,𝜎₄⟩, not included in ⟨𝜎₂,𝜎₅⟩ and containing at least one of Kuratowski operation 𝜎₂, 𝜎₃, 𝜎₄, or 𝜎₅. It will be discussed up to the isomorphism 𝔸. Obviously, ⟨𝜎₂⟩={𝜎₂} and ⟨𝜎₅⟩={𝜎₅} are groups.

7.1. Extensions of ⟨𝝈_𝟐⟩ and ⟨𝝈_𝟓⟩ with Elements of ⟨𝝈_𝟔,𝝈_𝟗⟩

Monoids ⟨𝜎₂,𝜎₆⟩={𝜎₂,𝜎₆,𝜎₁₀} and ⟨𝜎₂,𝜎₁₀⟩={𝜎₂,𝜎₁₀} have different numbers of elements. Also, ⟨𝜎₂,𝜎₆⟩ is isomorphic to ⟨𝜎₀,𝜎₆⟩. Nonisomorphic semigroups ⟨𝜎₂,𝜎₇⟩ and ⟨𝜎₂,𝜎₈⟩ are discussed above. The following three semigroups: (i)⟨𝜎₂,𝜎₁₁⟩=⟨𝜎₂,𝜎₆,𝜎₇⟩={𝜎₂,𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}, (ii)⟨𝜎₂,𝜎₁₂⟩=⟨𝜎₂,𝜎₆,𝜎₈⟩={𝜎₂,𝜎₆,𝜎₈,𝜎₁₀,𝜎₁₂}, (iii)⟨𝜎₂,𝜎₁₃⟩=⟨𝜎₂,𝜎₇,𝜎₈⟩={𝜎₂,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃} are not monoids. They are not isomorphic. Indeed, any isomorphism between these semigroups would lead an isomorphism between ⟨𝜎₆,𝜎₇⟩, ⟨𝜎₆,𝜎₈⟩, or ⟨𝜎₇,𝜎₈⟩. This is impossible, by Proposition 4 and because ⟨𝜎₇,𝜎₈⟩ consists of idempotents, but 𝜎₆ is not an idempotent. The nine-element semigroup on the set {𝜎₂,𝜎₆,𝜎₇,…,𝜎₁₃} is represented as ⟨𝜎₂,𝜎₉⟩. So, we have added four new semigroups, which are not isomorphic with the semigroups previously discussed. These are ⟨𝜎₂,𝜎₁₁⟩, ⟨𝜎₂,𝜎₁₂⟩, ⟨𝜎₂,𝜎₁₃⟩, and ⟨𝜎₂,𝜎₉⟩.

Using 𝔸, we have described eight semigroups—each one isomorphic to a semigroup previously discussed—which contains 𝜎₅ and elements (at least one) of ⟨𝜎₆,𝜎₉⟩. Collections of generators: ⟨𝜎₂,𝜎₆,𝜎₁₃⟩, ⟨𝜎₂,𝜎₇,𝜎₁₂⟩, ⟨𝜎₂,𝜎₈,𝜎₁₁⟩, ⟨𝜎₂,𝜎₁₁,𝜎₁₂⟩, ⟨𝜎₂,𝜎₁₁,𝜎₁₃⟩, and ⟨𝜎₂,𝜎₁₂,𝜎₁₃⟩ are minimal in ⟨𝜎₂,𝜎₉⟩. Also, collections of generators: ⟨𝜎₅,𝜎₇,𝜎₁₂⟩, ⟨𝜎₅,𝜎₈,𝜎₁₁⟩, ⟨𝜎₅,𝜎₉,𝜎₁₀⟩, ⟨𝜎₅,𝜎₁₀,𝜎₁₁⟩, ⟨𝜎₅,𝜎₁₀,𝜎₁₂⟩, and ⟨𝜎₅,𝜎₁₁,𝜎₁₂⟩ are minimal in ⟨𝜎₅,𝜎₆⟩.

7.2. Extensions of ⟨𝝈_𝟑⟩ and ⟨𝝈_𝟒⟩ by Elements from the Semigroup ⟨𝝈_𝟔,𝝈_𝟗⟩

Semigroups ⟨𝜎₃⟩={𝜎₃,𝜎₇,𝜎₁₁} and ⟨𝜎₄⟩={𝜎₄,𝜎₈,𝜎₁₂} are isomorphic by 𝔸. They are not monoids. The semigroup ⟨𝜎₃⟩ can be extended using elements of ⟨𝜎₆,𝜎₉⟩, in three following ways:(i)⟨𝜎₃,𝜎₆⟩=⟨𝜎₃,𝜎₁₀⟩={𝜎₃,𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}; (ii)⟨𝜎₃,𝜎₈⟩=⟨𝜎₃,𝜎₁₂⟩={𝜎₃,𝜎₆,𝜎₇,…,𝜎₁₃}; (iii)⟨𝜎₃,𝜎₉⟩=⟨𝜎₃,𝜎₁₃⟩={𝜎₃,𝜎₇,𝜎₉,𝜎₁₁,𝜎₁₃}.Semigroups ⟨𝜎₃,𝜎₆⟩ and ⟨𝜎₃,𝜎₉⟩ are not isomorphic. Indeed, suppose 𝐽∶⟨𝜎₃,𝜎₆⟩→⟨𝜎₃,𝜎₉⟩ is an isomorphism. Thus, 𝐽 is the identity on ⟨𝜎₃⟩, 𝐽(𝜎₆)=𝜎₉, and 𝐽(𝜎₁₀)=𝜎₁₃. This gives a contradiction, since 𝜎₃∘𝜎₆=𝜎₁₀ and 𝜎₃∘𝜎₉=𝜎₇.

Neither ⟨𝜎₃,𝜎₆⟩ nor ⟨𝜎₃,𝜎₉⟩ has a minimal collection of generators with three elements, so they give new isomorphism types. Also, ⟨𝜎₂,𝜎₉⟩ is not isomorphic to ⟨𝜎₃,𝜎₈⟩, since ⟨𝜎₂,𝜎₉⟩ has a unique pair of generators and ⟨𝜎₃,𝜎₈⟩=⟨𝜎₃,𝜎₁₂⟩.

Using 𝔸, we get—isomorphic to previously discussed ones—semigroups ⟨𝜎₄,𝜎₉⟩=⟨𝜎₄,𝜎₁₃⟩={𝜎₄,𝜎₈,𝜎₉𝜎₁₂,𝜎₁₃}, ⟨𝜎₄,𝜎₆⟩=⟨𝜎₄,𝜎₁₀⟩={𝜎₄,𝜎₆,𝜎₈𝜎₁₀,𝜎₁₂}, and ⟨𝜎₄,𝜎₇⟩=⟨𝜎₄,𝜎₁₁⟩={𝜎₄,𝜎₆,𝜎₇,…,𝜎₁₃}. There exist minimal collections of generators, such as follows: 𝜎₃,𝜎₈=𝜎₃,𝜎₆,𝜎₉=𝜎₃,𝜎₆,𝜎₁₃=𝜎₃,𝜎₉,𝜎₁₀=𝜎₃,𝜎₁₀,𝜎₁₃,𝜎₄,𝜎₇=𝜎₄,𝜎₆,𝜎₉=𝜎₄,𝜎₉,𝜎₁₀=𝜎₄,𝜎₆,𝜎₁₃=𝜎₄,𝜎₁₀,𝜎₁₃. (13)

7.3. More Generators from the Set {𝝈_𝟐,𝝈_𝟑,𝝈_𝟒,𝝈_𝟓}

Now we check that ⟨𝜎₂,𝜎₃⟩={𝜎₂,𝜎₃,𝜎₆,𝜎₇,𝜎₁₀𝜎₁₁} and ⟨𝜎₄,𝜎₅⟩={𝜎₄,𝜎₅,𝜎₈,𝜎₉,𝜎₁₂𝜎₁₃}=𝔸[⟨𝜎₂,𝜎₃⟩] and also ⟨𝜎₂,𝜎₄⟩={𝜎₂,𝜎₄,𝜎₆,𝜎₈,𝜎₁₀𝜎₁₂} and ⟨𝜎₃,𝜎₅⟩={𝜎₃,𝜎₅,𝜎₇,𝜎₉,𝜎₁₁𝜎₁₃}=𝔸[⟨𝜎₂,𝜎₄⟩] are two pairs of isomorphic semigroups which give two new isomorphism types. Each of these semigroups has six element, so in 𝕄 there are five six-element semigroups of three isomorphism types, since ⟨𝜎₂,𝜎₅⟩ has 6 elements which are idempotents.

In ⟨𝜎₃,𝜎₄⟩ there are seven semigroups which have three generators and have not two generators. These are (1)⟨𝜎₂,𝜎₃,𝜎₈⟩=⟨𝜎₂,𝜎₃,𝜎₉⟩=⟨𝜎₂,𝜎₃,𝜎₁₂⟩=⟨𝜎₂,𝜎₃,𝜎₁₃⟩; (2)⟨𝜎₄,𝜎₅,𝜎₆⟩=⟨𝜎₄,𝜎₅,𝜎₇⟩=⟨𝜎₄,𝜎₅,𝜎₁₀⟩=⟨𝜎₄,𝜎₅,𝜎₁₁⟩; (3)⟨𝜎₂,𝜎₄,𝜎₇⟩=⟨𝜎₂,𝜎₄,𝜎₉⟩=⟨𝜎₂,𝜎₄,𝜎₁₁⟩=⟨𝜎₂,𝜎₄,𝜎₁₃⟩; (4)⟨𝜎₃,𝜎₅,𝜎₆⟩=⟨𝜎₃,𝜎₅,𝜎₈⟩=⟨𝜎₃,𝜎₅,𝜎₁₀⟩=⟨𝜎₃,𝜎₅,𝜎₁₂⟩; (5)⟨𝜎₂,𝜎₅,𝜎₆⟩=⟨𝜎₂,𝜎₅,𝜎₉⟩=⟨𝜎₂,𝜎₅,𝜎₁₁⟩=⟨𝜎₂,𝜎₅,𝜎₁₂⟩; (6)⟨𝜎₂,𝜎₃,𝜎₅⟩; (7)⟨𝜎₂,𝜎₄,𝜎₅⟩.

8. Semigroups which Are Contained in ⟨𝝈_𝟑,𝝈_𝟒⟩

Groups ⟨𝜎₂⟩≅⟨𝜎₅⟩≅⟨𝜎₇⟩≅⟨𝜎₈⟩≅⟨𝜎₁₀⟩≅⟨𝜎₁₃⟩ have one element and are isomorphic.

Groups ⟨𝜎₆⟩≅⟨𝜎₉⟩≅⟨𝜎₁₁⟩≅⟨𝜎₁₂⟩, monoids ⟨𝜎₂,𝜎₁₀⟩≅⟨𝜎₅,𝜎₁₃⟩, and also semigroups ⟨𝜎₇,𝜎₁₀⟩≅⟨𝜎₈,𝜎₁₃⟩ and ⟨𝜎₇,𝜎₁₃⟩≅⟨𝜎₈,𝜎₁₀⟩ have two elements and Cayley tables as follows: 𝐴𝐵𝐴𝐵𝐵𝐴𝐴𝐵𝐴𝐵𝐴𝐴𝐵𝐵𝐵𝐴𝐴𝐵𝐴𝐵𝐴𝐵𝐴𝐵𝐴𝐵𝐴𝐴𝐴𝐵𝐵𝐵 (14)

Monoids ⟨𝜎₂,𝜎₆⟩≅⟨𝜎₅,𝜎₉⟩, and also semigroups ⟨𝜎₂,𝜎₇⟩≅⟨𝜎₅,𝜎₈⟩, ⟨𝜎₂,𝜎₈⟩≅⟨𝜎₅,𝜎₇⟩, and ⟨𝜎₃⟩≅⟨𝜎₄⟩ have three elements and Cayley tables as follows: 𝐴𝐵𝐶𝐴𝐵𝐴𝐵𝐶𝐶𝐵𝐶𝐵𝐶𝐵𝐶𝐴𝐵𝐶𝐴𝐴𝐵𝐶𝐵𝐶𝐵𝐶𝐶𝐶𝐵𝐶𝐴𝐵𝐶𝐴𝐵𝐴𝐶𝐶𝐶𝐵𝐵𝐵𝐶𝐶𝐶𝐴𝐵𝐶𝐴𝐵𝐶𝐵𝐵𝐶𝐵𝐶𝐶𝐵𝐶𝐵 (15)

Semigroups ⟨𝜎₆,𝜎₇⟩≅⟨𝜎₈,𝜎₉⟩, ⟨𝜎₆,𝜎₈⟩≅⟨𝜎₇,𝜎₉⟩, and ⟨𝜎₇,𝜎₈⟩ have four elements and Cayley tables as follows: 𝐴𝐵𝐶𝐷𝐴𝐵𝐶𝐷𝐴𝐵𝐶𝐴𝐵𝐶𝐷𝐷𝐴𝐵𝐶𝐷𝐶𝐷𝐴𝐵𝐴𝐵𝐶𝐷𝐴𝐶𝐴𝐴𝐶𝐵𝐷𝐵𝐵𝐷𝐶𝐴𝐶𝐶𝐴𝐷𝐵𝐷𝐷𝐵𝐴𝐵𝐶𝐷𝐴𝐵𝐴𝐶𝐶𝐴𝐶𝐷𝐵𝐵𝐷𝐷𝐴𝐶𝐶𝐴𝐷𝐵𝐵𝐷 (16)

Semigroups ⟨𝜎₂,𝜎₁₁⟩≅⟨𝜎₅,𝜎₁₂⟩, ⟨𝜎₂,𝜎₁₂⟩≅⟨𝜎₅,𝜎₁₁⟩, ⟨𝜎₂,𝜎₁₃⟩≅⟨𝜎₅,𝜎₁₀⟩, ⟨𝜎₃,𝜎₆⟩≅⟨𝜎₄,𝜎₉⟩, and ⟨𝜎₃,𝜎₉⟩≅⟨𝜎₄,𝜎₆⟩ have five elements. Semigroups ⟨𝜎₂,𝜎₃⟩≅⟨𝜎₄,𝜎₅⟩, ⟨𝜎₂,𝜎₄⟩≅⟨𝜎₃,𝜎₅⟩, and ⟨𝜎₂,𝜎₅⟩ have six elements. Construction of Cayley tables for these semigroups, as well as other semigroups, we leave it to the readers. One can prepare such tables removing from the Cayley table for 𝕄 some columns and rows. Then change Kuratowski operations onto letters of the alphabet. For example, ⟨𝜎₀,𝜎₂,𝜎₅⟩ has the following Cayley table: 0𝐴𝐵𝐶𝐷𝐸𝐹0𝐴0𝐴𝐵𝐶𝐷𝐸𝐹𝐵𝐴𝐴𝐶𝐶𝐸𝐸𝐶𝐶𝐵𝐷𝐵𝐹𝐷𝐷𝐹𝐷𝐶𝐸𝐶𝐶𝐸𝐸𝐶𝐸𝐷𝐷𝐹𝐹𝐷𝐷𝐹𝐹𝐸𝐸𝐶𝐶𝐸𝐸𝐶𝐹𝐷𝐹𝐹𝐷𝐷𝐹 (17) Preparing the Cayley table for ⟨𝜎₀,𝜎₂,𝜎₅⟩ we put 𝜎₀=0,𝜎₂=𝐴,𝜎₅=𝐵,𝜎₇=𝐶,𝜎₈=𝐷,𝜎₁₀=𝐸, and 𝜎₁₃=𝐹. This table immediately shows that semigroups ⟨𝜎₂,𝜎₅⟩ and ⟨𝜎₀,𝜎₂,𝜎₅⟩ have exactly two automorphisms. These are identities and restrictions of 𝔸.

Semigroup ⟨𝜎₆,𝜎₉⟩ has eight elements. Semigroups ⟨𝜎₂,𝜎₉⟩≅⟨𝜎₅,𝜎₆⟩ and ⟨𝜎₃,𝜎₈⟩≅⟨𝜎₄,𝜎₇⟩ have nine elements. Semigroups ⟨𝜎₂,𝜎₃,𝜎₈⟩≅⟨𝜎₄,𝜎₅,𝜎₇⟩, ⟨𝜎₂,𝜎₄,𝜎₇⟩≅⟨𝜎₃,𝜎₅,𝜎₈⟩, and ⟨𝜎₂,𝜎₅,𝜎₆⟩ have ten elements. Semigroups ⟨𝜎₂,𝜎₃,𝜎₅⟩≅⟨𝜎₂,𝜎₄,𝜎₅⟩ have eleven elements. In the end, the semigroup ⟨𝜎₃,𝜎₄⟩ has twelve elements.

Thus, the semigroup ⟨𝜎₃,𝜎₄⟩ includes fifty-seven semigroups, among which there are ten groups, fourteen monoids, and also forty-three semigroups which are not monoids. These semigroups consist of twenty-eight types of nonisomorphic semigroups, two non-isomorphism types of groups, two non-isomorphism types of monoids which are not groups, and twenty four non-isomorphism types of semigroups which are not monoids.

9. Viewing Semigroups Contained in 𝕄

9.1. Descriptive Data on Semigroups which Are Contained in 𝕄

There are one hundred eighteen, that is, 118=2⋅57+4, semigroups which are contained in 𝕄. These are fifty-seven semigroups contained in ⟨𝜎₃,𝜎₄⟩, fifty-seven monoids formed by adding 𝜎₀ to a semigroup contained in ⟨𝜎₃,𝜎₄⟩, groups ⟨𝜎₀⟩, ⟨𝜎₁⟩, and monoids ⟨𝜎₁,𝜎₆⟩=𝕄₁ and ⟨𝜎₁,𝜎₂⟩=𝕄.

There are fifty-six types of nonisomorphic semigroups in 𝕄. These are twenty-eight non-isomorphism types of semigroups contained in ⟨𝜎₃,𝜎₄⟩, twenty-six types of nonisomorphic monoids formed by adding 𝜎₀ to a semigroup contained in ⟨𝜎₃,𝜎₄⟩, and also 𝕄₁ and 𝕄. Indeed, adding 𝜎₀ to a semigroup which is not a monoid we obtain a monoid. In this way, we get twenty-four types of nonisomorphic monoids. Adding 𝜎₀ to a monoid contained in ⟨𝜎₃,𝜎₄⟩ we get two new non-isomorphism types of monoids. But, adding 𝜎₀ to a group contained in ⟨𝜎₃,𝜎₄⟩ we get no new type of monoid, since we get a monoid isomorphic to ⟨𝜎₂,𝜎₁₀⟩ or ⟨𝜎₂,𝜎₆⟩. The other two types are 𝕄₁ and 𝕄.

9.2. Semigroups which Are Not Monoids

Below we have reproduced, using the smallest number of generators and the dictionary order, a list of all 43 semigroups, which are included in the 𝕄 as follows:(1)⟨𝜎₂,𝜎₃⟩={𝜎₂,𝜎₃,𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}.(2)⟨𝜎₂,𝜎₃,𝜎₅⟩={𝜎₂,𝜎₃,𝜎₅,𝜎₆,…,𝜎₁₃}.(3)⟨𝜎₂,𝜎₃,𝜎₈⟩=⟨𝜎₂,𝜎₃,𝜎₉⟩=⟨𝜎₂,𝜎₃,𝜎₁₂⟩=⟨𝜎₂,𝜎₃,𝜎₁₃⟩={𝜎₂,𝜎₃,𝜎₆,𝜎₇,…,𝜎₁₃}. (4)⟨𝜎₂,𝜎₄⟩={𝜎₂,𝜎₄,𝜎₆,𝜎₈,𝜎₁₀,𝜎₁₂}.(5)⟨𝜎₂,𝜎₄,𝜎₅⟩={𝜎₂,𝜎₄,𝜎₅,…,𝜎₁₃}.(6)⟨𝜎₂,𝜎₄,𝜎₇⟩=⟨𝜎₂,𝜎₄,𝜎₉⟩=⟨𝜎₂,𝜎₄,𝜎₁₁⟩=⟨𝜎₂,𝜎₄,𝜎₁₃⟩={𝜎₂,𝜎₄,𝜎₆,𝜎₇…,𝜎₁₃}.(7)⟨𝜎₂,𝜎₅⟩={𝜎₂,𝜎₅,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃}.(8)⟨𝜎₂,𝜎₅,𝜎₆⟩=⟨𝜎₂,𝜎₅,𝜎₉⟩=⟨𝜎₂,𝜎₅,𝜎₁₁⟩=⟨𝜎₂,𝜎₅,𝜎₁₂⟩={𝜎₂,𝜎₅,𝜎₆,…,𝜎₁₃}.(9)⟨𝜎₂,𝜎₇⟩={𝜎₂,𝜎₇,𝜎₁₀}.(10)⟨𝜎₂,𝜎₈⟩={𝜎₂,𝜎₈,𝜎₁₀}.(11)⟨𝜎₂,𝜎₉⟩=⟨𝜎₂,𝜎₆,𝜎₁₃⟩=⟨𝜎₂,𝜎₇,𝜎₁₂⟩=⟨𝜎₂,𝜎₈,𝜎₁₁⟩=⟨𝜎₂,𝜎₁₁,𝜎₁₂⟩=⟨𝜎₂,𝜎₁₁,𝜎₁₃⟩  =  ⟨𝜎₂,𝜎₁₂,𝜎₁₃⟩={𝜎₂,𝜎₆,𝜎₇,…,𝜎₁₃}.(12)⟨𝜎₂,𝜎₁₁⟩=⟨𝜎₂,𝜎₆,𝜎₇⟩={𝜎₂,𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}.(13)⟨𝜎₂,𝜎₁₂⟩=⟨𝜎₂,𝜎₆,𝜎₈⟩={𝜎₂,𝜎₆,𝜎₈,𝜎₁₀,𝜎₁₂}.(14)⟨𝜎₂,𝜎₁₃⟩=⟨𝜎₂,𝜎₇,𝜎₈⟩={𝜎₂,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃}.(15)⟨𝜎₃⟩={𝜎₃,𝜎₇,𝜎₁₁}.(16)⟨𝜎₃,𝜎₄⟩={𝜎₂,𝜎₃,…,𝜎₁₃}.(17)⟨𝜎₃,𝜎₅⟩={𝜎₃,𝜎₅,𝜎₇,𝜎₉,𝜎₁₁,𝜎₁₃}.(18)⟨𝜎₃,𝜎₅,𝜎₆⟩=⟨𝜎₃,𝜎₅,𝜎₈⟩=⟨𝜎₃,𝜎₅,𝜎₁₀⟩=⟨𝜎₃,𝜎₅,𝜎₁₂⟩={𝜎₃,𝜎₅,𝜎₆,…,𝜎₁₃}.(19)⟨𝜎₃,𝜎₆⟩=⟨𝜎₃,𝜎₁₀⟩={𝜎₃,𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}.(20)⟨𝜎₃,𝜎₈⟩=⟨𝜎₃,𝜎₁₂⟩={𝜎₃,𝜎₆,𝜎₇,…,𝜎₁₃}.(21)⟨𝜎₃,𝜎₉⟩=⟨𝜎₃,𝜎₁₃⟩={𝜎₃,𝜎₇,𝜎₉,𝜎₁₁,𝜎₁₃}.(22)⟨𝜎₄⟩={𝜎₄,𝜎₈,𝜎₁₂}.(23)⟨𝜎₄,𝜎₅⟩={𝜎₄,𝜎₅,𝜎₈,𝜎₉,𝜎₁₂,𝜎₁₃}.(24)⟨𝜎₄,𝜎₅,𝜎₆⟩=⟨𝜎₄,𝜎₅,𝜎₇⟩=⟨𝜎₄,𝜎₅,𝜎₁₀⟩=⟨𝜎₄,𝜎₅,𝜎₁₁⟩={𝜎₄,𝜎₅,…,𝜎₁₃}.(25)⟨𝜎₄,𝜎₆⟩=⟨𝜎₄,𝜎₁₀⟩={𝜎₄,𝜎₆,𝜎₈,𝜎₁₀,𝜎₁₂}.(26)⟨𝜎₄,𝜎₇⟩=⟨𝜎₄,𝜎₁₁⟩={𝜎₄,𝜎₆,𝜎₇,…,𝜎₁₃}.(27)⟨𝜎₄,𝜎₉⟩=⟨𝜎₄,𝜎₁₃⟩={𝜎₄,𝜎₈,𝜎₉,𝜎₁₂,𝜎₁₃}.(28)⟨𝜎₅,𝜎₆⟩=⟨𝜎₅,𝜎₇,𝜎₁₂⟩=⟨𝜎₅,𝜎₈,𝜎₁₁⟩=⟨𝜎₅,𝜎₉,𝜎₁₀⟩=⟨𝜎₅,𝜎₁₀,𝜎₁₁⟩=⟨𝜎₅,𝜎₁₀,𝜎₁₂⟩  =  ⟨𝜎₅,𝜎₁₁,𝜎₁₂⟩={𝜎₅,𝜎₆,…,𝜎₁₃}.(29)⟨𝜎₅,𝜎₇⟩={𝜎₅,𝜎₇,𝜎₁₃}.(30)⟨𝜎₅,𝜎₈⟩={𝜎₅,𝜎₈,𝜎₁₃}.(31)⟨𝜎₅,𝜎₁₀⟩=⟨𝜎₅,𝜎₇,𝜎₈⟩={𝜎₅,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃}.(32)⟨𝜎₅,𝜎₁₁⟩=⟨𝜎₅,𝜎₇,𝜎₉⟩={𝜎₅,𝜎₇,𝜎₉,𝜎₁₁,𝜎₁₃}.(33)⟨𝜎₅,𝜎₁₂⟩=⟨𝜎₅,𝜎₈,𝜎₉⟩={𝜎₅,𝜎₈,𝜎₉,𝜎₁₂,𝜎₁₃}.(34)⟨𝜎₆,𝜎₇⟩=⟨𝜎₆,𝜎₁₁⟩=⟨𝜎₁₀,𝜎₁₁⟩={𝜎₆,𝜎₇,𝜎₁₀,𝜎₁₁}.(35)⟨𝜎₆,𝜎₈⟩=⟨𝜎₆,𝜎₁₂⟩=⟨𝜎₁₀,𝜎₁₂⟩={𝜎₆,𝜎₈,𝜎₁₀,𝜎₁₂}.(36)⟨𝜎₆,𝜎₉⟩=⟨𝜎₆,𝜎₁₃⟩=⟨𝜎₇,𝜎₁₂⟩=⟨𝜎₈,𝜎₁₁⟩=⟨𝜎₉,𝜎₁₀⟩=⟨𝜎₁₁,𝜎₁₂⟩  =  ⟨𝜎₆,𝜎₇,𝜎₈⟩=⟨𝜎₇,𝜎₈,𝜎₉⟩  =  ⟨𝜎₁₀,𝜎₁₁,𝜎₁₃⟩=⟨𝜎₁₀,𝜎₁₂,𝜎₁₃⟩={𝜎₆,𝜎₇,…,𝜎₁₃}.(37)⟨𝜎₇,𝜎₈⟩={𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃}.(38)⟨𝜎₇,𝜎₉⟩={𝜎₇,𝜎₉,𝜎₁₁,𝜎₁₃}.(39)⟨𝜎₇,𝜎₁₀⟩={𝜎₇,𝜎₁₀}.(40)⟨𝜎₇,𝜎₁₃⟩={𝜎₇,𝜎₁₃}.(41)⟨𝜎₈,𝜎₉⟩={𝜎₈,𝜎₉,𝜎₁₂,𝜎₁₃}.(42)⟨𝜎₈,𝜎₁₀⟩={𝜎₈,𝜎₁₀}.(43)⟨𝜎₈,𝜎₁₃⟩={𝜎₈,𝜎₁₃}.

9.3. Isomorphism Types of Semigroups Contained in 𝕄

Systematize the list of all isomorphism types of semigroups contained in the monoid 𝕄. Isomorphisms, which are restrictions of the isomorphism 𝔸, will be regarded as self-evident, and therefore they will not be commented on.(i)The monoid 𝕄 contains 12 groups with 2 isomorphism types. These are 7 one-element groups and 5 two-element groups.(ii)The monoid 𝕄 contains 8 two-element monoids with 2 isomorphism types. These are 𝜎₀ added to 6 one-element groups and ⟨𝜎₂,𝜎₁₀⟩≅⟨𝜎₅,𝜎₁₃⟩. Also, it contains 4 two-element semigroups, not monoids, with 2 isomorphism types. These are ⟨𝜎₇,𝜎₁₀⟩≅⟨𝜎₈,𝜎₁₃⟩ and ⟨𝜎₈,𝜎₁₀⟩≅⟨𝜎₇,𝜎₁₃⟩.(iii)The monoid 𝕄 contains 12 three-element monoids with 5 isomorphism types. These are 𝜎₀ added to 4 two-element groups and also ⟨𝜎₀,𝜎₂,𝜎₁₀⟩≅⟨𝜎₀,𝜎₅,𝜎₁₃⟩, ⟨𝜎₀,𝜎₇,𝜎₁₀⟩≅⟨𝜎₀,𝜎₈,𝜎₁₃⟩, ⟨𝜎₀,𝜎₈,𝜎₁₀⟩≅⟨𝜎₀,𝜎₇,𝜎₁₃⟩, and ⟨𝜎₂,𝜎₆⟩≅⟨𝜎₅,𝜎₉⟩.(iv)The monoid 𝕄 contains 6 three-element semigroups, not monoids, with 3 isomorphism types. These are ⟨𝜎₂,𝜎₇⟩≅⟨𝜎₅,𝜎₈⟩, ⟨𝜎₂,𝜎₈⟩≅⟨𝜎₅,𝜎₇⟩, and ⟨𝜎₃⟩≅⟨𝜎₄⟩.(v)The monoid 𝕄 contains 8 four-element monoids, each contains 𝜎₀, with 4 isomorphism types. These are semigroups from two preceding items that can be substantially extended by 𝜎₀.(vi)The monoid 𝕄 contains 5 four-element semigroups, not monoids, with 3 isomorphism types. These are ⟨𝜎₆,𝜎₇⟩≅⟨𝜎₈,𝜎₉⟩, ⟨𝜎₆,𝜎₈⟩≅⟨𝜎₇,𝜎₉⟩ and ⟨𝜎₇,𝜎₈⟩. These semigroups extended by 𝜎₀ yield 5 monoids, all which consist of five elements, with 3 isomorphism types.(vii)The monoid 𝕄 contains 10 five-element semigroups—not monoids, with 5 isomorphism types. These are ⟨𝜎₂,𝜎₁₁⟩≅⟨𝜎₅,𝜎₁₂⟩, ⟨𝜎₂,𝜎₁₂⟩≅⟨𝜎₅,𝜎₁₁⟩, ⟨𝜎₂,𝜎₁₃⟩≅⟨𝜎₅,𝜎₁₀⟩, ⟨𝜎₃,𝜎₆⟩≅⟨𝜎₄,𝜎₉⟩, and ⟨𝜎₃,𝜎₉⟩≅⟨𝜎₄,𝜎₆⟩. These semigroups extended by 𝜎₀ yield 10 monoids, all of which consist of six elements, with 5 isomorphism types. We get 10 new isomorphism types, since the semigroups are distinguished by semigroups ⟨𝜎₃⟩ and ⟨𝜎₄⟩, and by nonisomorphic semigroups ⟨𝜎₆,𝜎₇⟩, ⟨𝜎₆,𝜎₈⟩, and ⟨𝜎₇,𝜎₈⟩.(viii)The monoid 𝕄 contains 5 six-element semigroups, not monoids, with 3 isomorphism types. These are ⟨𝜎₂,𝜎₃⟩≅⟨𝜎₄,𝜎₅⟩, ⟨𝜎₂,𝜎₄⟩≅⟨𝜎₃,𝜎₅⟩, and ⟨𝜎₂,𝜎₅⟩. These semigroups extended by 𝜎₀ yield 5 monoids, all of which consist of seven elements, with 3 isomorphism types. We get 6 new isomorphism types, since the semigroups are distinguished by non isomorphic semigroups ⟨𝜎₆,𝜎₇⟩, ⟨𝜎₆,𝜎₈⟩, and ⟨𝜎₇,𝜎₈⟩.(ix)The monoid 𝕄 contains no seven-element semigroup, not a monoid, no eight-element monoid and the only semigroup ⟨𝜎₆,𝜎₉⟩ with exactly eight elements and the only monoid ⟨𝜎₀,𝜎₆,𝜎₉⟩ with exactly nine elements.(x)The monoid 𝕄 contains 4 nine-element semigroups, not monoids, with 2 isomorphism types. These are ⟨𝜎₂,𝜎₉⟩≅⟨𝜎₅,𝜎₆⟩ and ⟨𝜎₃,𝜎₈⟩≅⟨𝜎₄,𝜎₇⟩. The semigroup ⟨𝜎₂,𝜎₉⟩ does not contain a semigroup isomorphic to ⟨𝜎₃⟩, hence it is not isomorphic to ⟨𝜎₃,𝜎₈⟩. These semigroups extended by 𝜎₀ yield 4 monoids, all of which consist of ten elements, with 2 isomorphism types.(xi)The monoid 𝕄 contains 6 ten-element semigroups, not monoids, with 4 isomorphism types. These are ⟨𝜎₁,𝜎₆⟩, ⟨𝜎₂,𝜎₃,𝜎₈⟩≅⟨𝜎₄,𝜎₅,𝜎₆⟩, ⟨𝜎₂,𝜎₄,𝜎₇⟩≅⟨𝜎₃,𝜎₅,𝜎₆⟩, and ⟨𝜎₂,𝜎₅,𝜎₆⟩. These semigroups (except ⟨𝜎₁,𝜎₆⟩) extended by 𝜎₀ yield 5 monoids, all of which consist of ten elements, with 3 isomorphism types. We get 6 new isomorphism types, since the semigroups are distinguished by not isomorphic semigroups ⟨𝜎₂,𝜎₃⟩, ⟨𝜎₂,𝜎₄⟩, and ⟨𝜎₂,𝜎₅⟩.(xii)The monoid 𝕄 contains 2 isomorphic semigroups, not monoids, which consist of 11 elements, that is, ⟨𝜎₂,𝜎₃,𝜎₅⟩≅⟨𝜎₂,𝜎₄,𝜎₅⟩. These semigroups extended by 𝜎₀ yield 2 isomorphic monoids, which consist of 12 elements. The monoid 𝕄 contains no larger semigroup with the exception of itself.

10. Cancellation Rules Motivated by Some Topological Properties

10.1. Some Consequences of the Axiom ∅=∅⁻

So far, we used only the following relations (above named cancellation rules): 𝜎₂∘𝜎₂=𝜎₂, 𝜎₁∘𝜎₁=𝜎₀, 𝜎₂∘𝜎₁₂=𝜎₆, and 𝜎₂∘𝜎₁₃=𝜎₇. When one assumes 𝑋≠∅=∅⁻, then 𝑋=𝜎₀(𝑋)=𝜎₂(𝑋)=𝜎₅(𝑋)=𝜎₇(𝑋)=𝜎₈(𝑋)=𝜎₁₀(𝑋)=𝜎₁₃(𝑋),∅=𝜎₁(𝑋)=𝜎₃(𝑋)=𝜎₄(𝑋)=𝜎₆(𝑋)=𝜎₉(𝑋)=𝜎₁₁(𝑋)=𝜎₁₂(𝑋). (18) Using the substitution 𝐴↦𝐴^𝑐, one obtains equivalent relations between operations from the set {𝜎₁,𝜎₃,𝜎₄,𝜎₆,𝜎₉,𝜎₁₁,𝜎₁₂}, and conversely. Therefore, cancellation rules are topologically reasonable only between the operations from the monoid as follows: 𝜎₀,𝜎₂,𝜎₅=𝜎₀,𝜎₂,𝜎₅,𝜎₇,𝜎₈,𝜎₁₀,𝜎₁₃. (19) Chapman, see [3], considered properties of subsets with respect to such relations. Below, we are going to identify relations that are determined by some topological spaces; compare [10, 11].

10.2. The Relation 𝝈_𝟎=𝝈_𝟐

If a topological space 𝑋 is discrete, then there exist two Kuratowski operation, only. These are 𝜎₀ and 𝜎₁. So, the monoid of Kuratowski operations reduced to the group ⟨𝜎₁⟩.

The relation 𝜎₀=𝜎₂ is equivalent to any relation 𝜎₀=𝜎_𝑖, where 𝑖∈{5,7,8,10,13}. Any such relation implies that every subset of 𝑋 has to be closed and open; that is, 𝑋 has to be discrete. However, one can check these using (only) the facts that 𝜎₁ is an involution and 𝜎₂ is an idempotent and the cancellation rules, that is, the Cayley table for 𝕄. So, 𝜎₀=𝜎₂ follows 𝜎₀=𝜎₂=𝜎₅=𝜎₇=𝜎₈=𝜎₁₀=𝜎₁₃. (20)

10.3. The Relation 𝝈_𝟐=𝝈_𝟓

Topologically, 𝜎₂=𝜎₅ means that 𝑋 must be discrete. This is so because 𝐴^𝑐−𝑐⊆𝐴⊆𝐴⁻ for any 𝐴⊆𝑋.

10.4. The Relation 𝝈_𝟐=𝝈_𝟕

Topologically, the relation 𝜎₂=𝜎₇ implies 𝜎₀=𝜎₂. But it requires the use of topology axioms ∅=∅⁻ and 𝐶⁻∪𝐵⁻=(𝐶∪𝐵)⁻ for each 𝐶 and 𝐵.

Lemma 5. For any topological space 𝜎₂=𝜎₇ implies 𝜎₂=𝜎₈.

Proof. If 𝜎₂=𝜎₇, then 𝐴≠∅⇒𝐴^𝑐−𝑐≠∅, for any 𝐴⊆𝑋. Indeed, if 𝐴^𝑐−𝑐=∅, then 𝜎₇(𝐴)=∅⁻=∅. Since 𝐴≠∅, then 𝜎₂(𝐴)≠∅. Hence 𝜎₂(𝐴)≠𝜎₇(𝐴), a contradiction.
The axiom 𝐶⁻∪𝐵⁻=(𝐶∪𝐵)⁻ implies that always 𝐴⁻∩𝐴^{−𝑐−𝑐−𝑐}=∅. (21) Thus, the additional assumption 𝜎₂=𝜎₇ follows that always 𝐴⁻∩𝐴^−𝑐−=∅. Therefore, always 𝐴^−𝑐−=𝐴^−𝑐, but this means that any closed set has to be open; in other words, 𝜎₂=𝜎₈.

Proposition 6. For any topological space 𝜎₂=𝜎₇ implies 𝜎₀=𝜎₂.

Proof. But the relation 𝜎₂=𝜎₇ is equivalent with 𝜎₅=𝜎₈. By Lemma 5, we get 𝜎₂=𝜎₅. Finally 𝜎₀=𝜎₂.

The relation 𝜎₂=𝜎₇ has interpretation without the axiom ∅=∅⁻. Indeed, suppose 𝑋={𝑎,𝑏}. Put 𝜎₂(∅)={𝑎}=𝜎₂({𝑎}),𝑋=𝜎₂(𝑋)=𝜎₂({𝑏}). (22) Then, check that 𝜎₂=𝜎₇ and 𝜎₈(∅)=𝜎₅({𝑎})=𝜎₄({𝑏})=𝜎₁(𝑋)=∅; (23) in other words, 𝜎₂=𝜎₇ and 𝜎₂≠𝜎₈. However, 𝜎₂=𝜎₇ is equivalent to 𝜎₅=𝜎₈.

This relation implies 𝜎₂=𝜎₇=𝜎₁₀, 𝜎₅=𝜎₈=𝜎₁₃, 𝜎₃=𝜎₆=𝜎₁₁, and 𝜎₄=𝜎₉=𝜎₁₂. For this interpretation, the monoid 𝕄/(𝜎₂=𝜎₇), consisting of Kuratowski operation over a such 𝑋, has six elements, only. In 𝕄/(𝜎₂=𝜎₇), there are relations covered by the following proposition, only.

Proposition 7. For any monoid with the Cayley table as for 𝕄, the relation 𝜎₂=𝜎₇ implies (i)𝜎₂=𝜎₇=𝜎₁₀=𝜎₇∘𝜎₂; (ii)𝜎₅=𝜎₁∘𝜎₂∘𝜎₁=𝜎₁∘𝜎₇∘𝜎₁=𝜎₈=𝜎₅∘𝜎₂=𝜎₅∘𝜎₇=𝜎₁₃; (iii)𝜎₃=𝜎₂∘𝜎₁=𝜎₇∘𝜎₁=𝜎₆=𝜎₁₀∘𝜎₁=𝜎₁₁; (iv)𝜎₄=𝜎₁∘𝜎₂=𝜎₁∘𝜎₇=𝜎₉=𝜎₁∘𝜎₁₀=𝜎₁₂.

Thus, the Cayley table does not contain the complete information resulting from the axioms of topology.

10.5. The Relation 𝝈_𝟐=𝝈_𝟖

Topologically, the relation 𝜎₂=𝜎₈ means that any closed set is open, too. Thus, if 𝑋={𝑎,𝑏,𝑐} is a topological space with the open sets 𝑋, ∅, {𝑎,𝑏}, and {𝑐}, then 𝕄/(𝜎₂=𝜎₈) is the monoid of all Kuratowski operations over 𝑋. Relations 𝜎₂=𝜎₈ and 𝜎₅=𝜎₇ are equivalent. They imply relations: 𝜎₂=𝜎₈=𝜎₁₀, 𝜎₅=𝜎₇=𝜎₁₃, 𝜎₃=𝜎₉=𝜎₁₁, and 𝜎₄=𝜎₆=𝜎₁₂. The permutation 𝜎₀𝜎₁𝜎₂𝜎₃𝜎₄𝜎₅𝜎₀𝜎₁𝜎₅𝜎₄𝜎₃𝜎₂ (24) determines the isomorphism between monoids 𝕄/(𝜎₂=𝜎₇) and 𝕄/(𝜎₂=𝜎₈).

10.6. The Relations 𝝈_𝟐=𝝈_𝟏𝟎 and 𝝈_𝟐=𝝈_𝟏𝟑

Topologically, the relation 𝜎₂=𝜎₈ means that any nonempty closed set has nonempty interior. For each 𝐴, the closed set 𝐴⁻∩𝐴^−𝑐− has empty interior, so the relation 𝜎₂=𝜎₁₀ implies that 𝐴^−𝑐 is closed. Hence, any open set has to be closed, so it implies 𝜎₂=𝜎₈. The relation 𝜎₂=𝜎₁₃ follows that each closed set has to be open, so it implies 𝜎₂=𝜎₈, too.

10.7. The Relation 𝝈_𝟕=𝝈_𝟖

Using the Cayley table for 𝕄, one can check that the relations 𝜎₇=𝜎₈ and 𝜎₁₀=𝜎₁₃ are equivalent. Each of them gives 𝜎₇=𝜎₈=𝜎₁₀=𝜎₁₃ and 𝜎₆=𝜎₉=𝜎₁₁=𝜎₁₂. If 𝑋={𝑎,𝑏} is a topological space with the open sets 𝑋, ∅, and {𝑎}, then 𝕄/(𝜎₇=𝜎₈) is the monoid of all Kuratowski operations over 𝑋 and consists of 8 elements.

10.8. The Relation 𝝈_𝟕=𝝈_𝟏𝟎

Using the Cayley table for 𝕄, one can check that the relations 𝜎₇=𝜎₁₀, 𝜎₈=𝜎₁₃, 𝜎₆=𝜎₁₁, and 𝜎₉=𝜎₁₂ are equivalent. If 𝑋 is a sequence converging to the point 𝑔 and 𝑔∈𝑋, then 𝕄/(𝜎₇=𝜎₁₀) is the monoid of all Kuratowski operations over 𝑋 and consists of 10 elements.

10.9. The Relation 𝝈_𝟕=𝝈_𝟏𝟑

Using the Cayley table for 𝕄, one can check that the relations 𝜎₇=𝜎₁₃ and 𝜎₈=𝜎₁₀ are equivalent. Also 𝜎₆=𝜎₁₂ and 𝜎₉=𝜎₁₁. These relations give the monoid with the following Cayley table, where the row and column marked by the identity are omitted: 𝜎₁𝜎₂𝜎₃𝜎₄𝜎₅𝜎₆𝜎₇𝜎₈𝜎₉𝜎₁𝜎₀𝜎₄𝜎₅𝜎₂𝜎₃𝜎₈𝜎₉𝜎₆𝜎₇𝜎₂𝜎₃𝜎₂𝜎₃𝜎₆𝜎₇𝜎₆𝜎₇𝜎₈𝜎₉𝜎₃𝜎₂𝜎₆𝜎₇𝜎₂𝜎₃𝜎₈𝜎₉𝜎₆𝜎₇𝜎₄𝜎₅𝜎₄𝜎₅𝜎₈𝜎₉𝜎₈𝜎₉𝜎₆𝜎₇𝜎₅𝜎₄𝜎₈𝜎₉𝜎₄𝜎₅𝜎₆𝜎₇𝜎₈𝜎₉𝜎₆𝜎₇𝜎₆𝜎₇𝜎₈𝜎₉𝜎₈𝜎₉𝜎₆𝜎₇𝜎₇𝜎₆𝜎₈𝜎₉𝜎₆𝜎₇𝜎₆𝜎₇𝜎₈𝜎₉𝜎₈𝜎₉𝜎₈𝜎₉𝜎₆𝜎₇𝜎₆𝜎₇𝜎₈𝜎₉𝜎₉𝜎₈𝜎₆𝜎₇𝜎₈𝜎₉𝜎₈𝜎₉𝜎₆𝜎₇ (25)

If a space 𝑋 is extremally disconnected, then the closures of open sets are open; compare [2, page 452]. It follows that 𝜎₆=𝜎₁₂. The space 𝑋={𝑎,𝑏} with the open sets 𝑋, ∅, and {𝑎} is extremally disconnected. But it contains a one-element open and dense set {𝑎}, and it follows that 𝜎₇=𝜎₈. Similar is for the space 𝛽𝑁; see [2, pages 228 and 453] to find the definition and properties of 𝛽𝑁. There are Hausdorff extremally disconnected spaces which are dense in itself. For example, the Stone space of the complete Boolean algebra of all regular closed subsets of the unit interval, compare [12]. For such spaces 𝜎₇≠𝜎₈ and 𝜎₇=𝜎₁₃. To see this, suppose a Hausdorff 𝑋 is extremally disconnected and dense in itself. Let 𝑋=𝑈∪𝑉∪𝑊, where sets 𝑈,𝑉, and 𝑊 are closed and open. Consider a set 𝐴=𝐴^𝑐−𝑐∪𝐵∪𝐶, such that (i)𝐶^𝑐−𝑐=∅ and 𝐶⁻=𝑊; (ii)∅≠𝐵⊆𝑉 and 𝐵^{−𝑐−𝑐}=∅; (iii)𝑈=𝐴^{𝑐−𝑐−}≠𝐴^𝑐−𝑐. Then check that (i)𝜎₀(𝐴)=𝐴 and 𝜎₁(𝐴)=𝑋⧵(𝐴^𝑐−𝑐∪𝐵∪𝐶); (ii)𝜎₂(𝐴)=𝑈∪𝐵⁻∪𝑊 and 𝜎₃(𝐴)=𝑋⧵𝐴^𝑐−𝑐; (iii)𝜎₄(𝐴)=𝑉⧵𝐵⁻ and 𝜎₅(𝐴)=𝐴^𝑐−𝑐; (iv)𝜎₆(𝐴)=𝜎₁₂(𝐴)=𝑉 and 𝜎₇(𝐴)=𝜎₁₃(𝐴)=𝑈; (v)𝜎₈(𝐴)=𝜎₁₀(𝐴)=𝑈∪𝑊 and 𝜎₉(𝐴)=𝜎₁₁(𝐴)=𝑉∪𝑊. Hence we have that 𝜎₇≠𝜎₈. Note that, if 𝑊=∅, then 𝜎₇(𝐴)=𝜎₈(𝐴). This is the case of subsets of 𝛽𝑁.