Artículo Original

A CLASS OF CONGRUENCIES ON DISTRIBUTIVE SEMILATTICE

UNA CLASE DE CONGRUENCIAS EN SEMIRRETÍCULO DISTRIBUTIVO

Dekeba, Tolesa; Reta, Tesfu

Tolesa Dekeba

toleyoobii2020@gmail.com

Dire Dawa University Etiopía

Tesfu Reta

tesfur2013@gmail.com

Dire Dawa University Etiopía

Revista de Investigaciones Universidad del Quindío

Universidad del Quindío, Colombia

ISSN: 1794-631X

ISSN-e: 2500-5782

Periodicity: Anual

vol. 34, no. 1, 2022

riuq@uniquindio.edu.co

Received: 25 September 2021

Accepted: 10 February 2022

URL: http://portal.amelica.org/ameli/journal/517/51731105173110025/

Non-profit publishing model to preserve the academic and open nature of scientific communication

ePub generated from XML JATS4R

Abstract: In this paper we, contribute the notation of natural epimorphism of a semilattice on the quotient semilattice and subsemilattice. If S is distributive semilattice and F is a filter of S, then we demonstrate that θF is the smallest congruence on S containing F in a single equivalence class and that S/θF is distributive. In addition, the author proved that map F↦θF is an isomorphism from the lattice of F0(S) all non-empty filters of S into a permutable sublattice of the lattice C(S) of all congruencies on S.

Keywords: Congruence class of semilattice, Distributive Semilattice, Natural epimorphism of semilattice, Quotient semilattice.

Resumen: En este trabajo contribuimos con la notación del epimorfismo natural de una semirredura sobre el cociente semirreticulado y subsemretículo. Si S es una semirrejilla distributiva y F es un filtro de S, entonces demostramos que θF es la congruencia más pequeña en S que contiene F en una sola clase de equivalencia y que S/θF es distributiva. Además, el autor demostró que el mapa FI↦F es un isomorfismo de la red F0(S) de todos los filtros no vacíos de S en una subred permutable de la red C(S) de todas las congruencias en S.

Palabras clave: clase de congruencia de semirredura, Semirretículo distributivo, Epimorfismo natural de semirreduras, Cociente de semirreduras.

INTRODUCTION

In [14] (Klein Barmen 1939) introduced the concept of semilattice [13]. A semilattice is a partially ordered set in which any two elements have a greatest lower bound, but not necessarily upper bound. The class of distributive semilattices is a significant subclass of semilattices. Then, several authors were studied the class of distributive semilattices. The author refer the reader to [1], [2], [3], [4], [7], [9], [11], [17], [18], [19], and [21] for distributive semilattices. The concept of 0-distributive semilattices is another important extension of the class of distributive semilattices. In [19] (J.C.Varlet 1968) first introduced the concept of 0-distributive lattices. Then many authors including [3], [6], [10], [15], [16], and [20] studied concept of 0-distributive for lattices and semilattices.

Let S be a semilattice. A non-empty subset L of S is called directed above if, for any a,b ϵ L there exists c ϵ L such that a ≤ c and b ≤ c. L is said to be a final segment if, for any a L and x ϵ S, a ≤ x implies x ϵ L. In addition, a non-empty subset L is called directed below if, for any, a,b ϵ L, there exists c ϵ L such that c ≤ a and c ≤ b. L is called an initial segment if, for any a ϵ I and x ϵ S, x ≤ a implies x ϵ I. Following this concept, many authors developed the concept of filters and ideals of semilattice. The we refer the readers to [7], [16], [17] , and [18].

Let S and L be semilattices. A map f : S → L is said to be a homomorphism, if f is join preserving and meet preserving, that is, for all a,b ϵ S.

A bijective homomorphism is a semilattice isomorphism [22]. If f : L → K is a one-to-one homomorphism, then the sub-semilattice of f(S} L is isomorphic to S and we refer to f as an embedding (of S into L). In [18] (Swamy 1979) introduced the class of natural epimorphism of a semilattice on the quotient semilattice, and he denoted the natural epimorphism of semilattice S on the quotient semilatticeby S/θF by πF. For any sub-semilattice L of a semilattice S, define

After (Swamy 1979) nobody gives attention to the concept of congruence classes on the distributive semilattice as far as we investigate it. Therefore, by following this, we consider and demonstrate that; map F↦θF is an isomorphism from the lattice F0(S) of the class of all non-empty filters of S into a permutable sublattice of the lattice C(S) of the classes of all congruencies on S. In addition, we prove that θF is the smallest congruence on S containing F in a single equivalence class and that S/θL is distributive.

The manuscript is structured as follows. In this section, we introduced the background of semilattice with basic results about semilattices which are already exists and needed in developing this manuscript. In Section 2, we introduced the concepts of distributive semilattices. Some important Theorems and Corollaries in distributive semilattices and pseudo-complemented semilattice were also presented in this section. In section 3, we will prove the theorems belonging to classes of congruence in distributive semilattice.

PRELIMINARIES

In this section, the author will present some necessary notations and definitions.

Definition: 2.1. [2] Let S be a semilattice. A non-empty subset F of S is called a filter if;

Definition: 2.2. [3] A non-empty subset I of S is called an ideal if;

A filter (ideal) F of a semilattice S is called a proper filter (ideal) if ≠ S. A maximal filter (ideal) F of S is a proper filter (ideal), which is not contained in any other proper filter (ideal), that is, if there is a proper filter (ideal) G such that F ⊆ G, then F = G. (see [4])

Let F(S) and I(S) denotes the set of all filters of S and the set of all ideals of S respectively. Then, for any a ϵ S, let (a] denote the ideal

{x ϵ S : x ≤ a} and [a) denotes the filter

{x ϵ S : a ≤ x}. (see[11])

Definition: 2.3. [5] A semilattice S is said to be distributive if, for any a,b,c ϵ S, such that a Λ b ≤ c, there exist x, y ϵ S such that a ≤ x, b ≤ y, and x Λ y = c.

Lemma 2.4. [4] If S is a distributive semilattice, S is directed above.

Proof. Suppose S be a distributive semilattice, then for any a,b ϵ S, a Λ b ϵ S and a Λ b ≤ b. There exists x,y ϵ S such that a ≤ x, b ≤ y, and x Λ y = b. Also b = x Λ y ≤ x . Therefore, for any a,b ϵ S, there exists x ϵ S such that a ≤ x and b ≤ x.

Hence S is directed above.

Theorem 2.5. [6] Let I be an ideal of S and J a ﬁlter of S such that I ∩ J ≠ Ø. Then, I ∩ J is a distributive sub-semilattice of S.

Proof. Suppose I be an ideal of S and J a ﬁlter of S such that I ∩ J ≠ Ø. Let a,b ϵ I ∩ J, then a Λ b ϵ I. Since I is the initial segment, there exist c J ϵ such that c ≤ a and b ≤ c.

This implies c ≤ a Λ b and a Λ b ϵ J since J is the ﬁnal segment. Therefore I ∩ J is a sub-semilattice of S.

Let a,b,c ϵ I ∩ J such that a Λ b ≤ c. There exists x,y ϵ S such that a ≤ x, b ≤ y, and x Λ y = c. Since I is an ideal of S, there exist z ϵ I such that a ≤ z, b ≤ z and c ≤ z.

Now c = c Λ z = (x Λ z) Λ (y Λ z), a ≤ x Λ z ϵ I ∩ J and b ≤ y Λ z ϵ I ∩ J. Therefore I ∩ J is distributive sub-semilattice.

Definition 2.6. [19] A semilattice S with 0 is called 0 - distributive if for any a,b,c ϵ S such that a Λ b = 0 = a Λ c implies a Λ d = 0 for some d ≥ b,c.

Proposition 2.7. [16] Let S be distributive semilattice. Let I be an ideal of S and J a filter of S such that I ∩ J ≠ Ø then, I ∩ J is a distributive sub-semilattice of S.

Proof: Suppose that I is an ideal of S and J is a filter of S such that I ∩ J ≠ Ø. Let a,b ϵ I ∩ J, then a Λ b ϵ I. Since I is the initial segment, there exist c ϵ J such that c≤ a and b≤ c. Hence c ≤ a Λ b and a Λ b ϵ J, since J is the final segment. Thus I ∩ J is a sub-semilattice of S. Let a,b, and c ϵ I ∩ J such that a Λ b ≤ c. There exists x and y ϵ S such that a ≤ x, b ≤ y and x Λ y = c. Since I is an ideal of S, there exist z ϵ I such that a≤z, b≤z, and c≤z. Then we have c = c Λ z = (x Λ z) Λ (y Λ z), a ≤ x Λ z ϵ I ∩ J and b ≤ y Λ z ϵ I ∩ J. Hence I ∩ J is distributive.

Theorem 2.8. [18] Let (S, Λ, V) be a lattice. Then the following are equivalent:

1. (S, Λ, V) is a distributive lattice.

2. (S, Λ) is a distributive meet semilattice.

3. (S, V) is a distributive join semilattice.

Corollary 2.9. [18] Every non-empty ideal (filter) of S is a distributive sub-semilattice of S.

Proof: Suppose that I is a non-empty ideal of S. Let a,b ϵ I such that a Λ b ≤ c. There exist x,y ϵ S such that a ≤ x and b ≤ y, and c=cΛy as S is distributive. There exist z ϵ I such that a≤z, b≤z and c≤z as I is an ideal of S.

Now. c = c Λ z = (x Λ y) Λ z = (x Λ z) Λ (y Λ z)ϵ I

Hence I is distributive.

Similarly, suppose that J is a nonempty filter of S. For any a,b ϵ J, there exist c ϵ J such that c≤ a and

as J is the final segment. Then, J is a sub-semilattice of S. Let a,b,c ϵ J such that a Λ b ≤ c, there exist x,y ϵ S such that a ≤ x, b ≤ y, and x Λ y = c.

Now as a,b and c ϵ J, then x,y ϵ J. Hence xΛy=cϵJ.

Therefore J is distributive.

Theorem 2.9. Let I be an ideal of S and L is a sub-semilattice of S such that I ∩ L = Ø, then, there exists a prime ideal P of S such that I C P and P ∩ L = Ø.

Proof: Suppose that T ={xϵS : a ≤ x for some aϵL}. Then T is the filter of S and T∩I=Ø and by Zorn’s lemma, there exists a filter F of S which is maximal among all filters containing L, and disjoint from I. Now I C S - F and (S-F)∩L=Ø.

Let us prove F is a prime filter. Let x,y ϵ S- F, then by maximalist of F there exists

There exist c ϵ I such that a ≤ c and b ≤ c

Therefore F is a prime filter of S.

Theorem 2.10. [21] Every maximal ideal (filter) of S is prime.

Proof: Let S be distributive semilattice and Q is a maximal filter of S. Then S-Q is minimal ideal. Then there exist a prime ideal I such that S-Q⊆I then Q ∩ I = Ø

Let x,y∉Q, then x,y ϵ S-Q. Then there exist aϵ I ∩ (Q Λ[x)) and b ϵ I ∩ (Q Λ[y)).

This implies there exist c ϵ I such that a≤ c and b≤ c, since I directed above.

Hence

Therefore Q is a prime filter of S. Hence the result.

Theorem 2.11. [17] Let S be a semilattice. S has a largest element if and only if the intersection of all nonempty filters is again non-empty.

Proof: Suppose that S has, a largest element says t, then t ϵ ∩FϵF(s) F. Conversely suppose that the intersection of all nonempty filters (ideals) is again non-empty, by proposition 2.3, S has a Largest (smallest) element.

Lemma 2.12. [17] Let P C S. Then P is a prime filter of S if and only if S - P is a prime ideal.

Proof: Suppose P is a prime filter of S, it is non-empty and proper. To show S-P is directed above, let a,b∉S-P. Suppose if possible [a)∩[b)∩S-P=Ø, then [a)∩[b)⊆P. This implies [a)⊆P or [b)⊆P. Since P is prime then a ∉ S - P or b ∉ S - P. This is a contradiction. Therefore ([a)∩[b))∩S-P≠ Ø

So S - P is an ideal. To show that S - P is prime, suppose

Hence a ϵ S - P or b ϵ S - P.

Conversely, suppose S - P be a prime ideal, then S - P ≠ Ø, S. So that P ≠ Ø, S. Since S - P is directed above, and then P is directed below. If a,b ϵ P then a,b ∉ S - P. Since S - P is prime, a Λ b ∉ S - P. Thus a Λ b ϵ P, and so P is a filter.

Now, to show that P is a prime filter, we suppose if possible Q∩R⊆P and

and R∩S-P≠Ø, then there exists a ϵ Q ∩ S - P and

with

This is a contradiction.

Therefore

is prime.

CONGRUENCIES ON DISTRIBUTIVE SEMILATTICE

Definition 3.1. Let S be a semilattice. A homomorphism between two meet semilattices S and T is a map f : S → T with the property that f (a Λ b) = f (a) Λ f (b), where a,b ϵ S. A semilattice homomorphism is called semilattice isomorphism

Theorem 3.2. Let S be any semilattice, which is directed above. Then S is distributive if and only if S/θl is distributive for every subsemilattice L of S.

Proof: Suppose S be a distributive semilattice and L a subsemilattice of S. Let x,y,z ϵ S such that

such that x Λ y Λ a = x Λ y Λ z Λ a.

such that x Λ a ≤ b, y Λ a ≤ c and b Λ c = z

Now, πL (x Λ a) ≤ πL (b), πL (y Λ a) ≤ πL (c) and πL (b) Λ πL (c) = πL (z). Therefore S/θl is distributive for every subsemilattice L of S.

Conversely, let a,b,and c ϵ S such that aΛb≤c. Choose d ϵ S such that a,b, and c ϵ (d] and put L = (d]. Since S/θl is distributive, there exists e, f ϵ S suth that πL (a) ≤ πL (e), πL (b) ≤ πL (f) and πL (e) Λ πL (f) = πL (c) x,y and z ϵ F such that e Λ f Λ x = c Λ x, e Λ a Λ y = a Λ y and f Λ b Λ z = b Λ z which implies that e Λ f Λ x = c, e Λ a =a and f Λ b = b. Therefore and (e Λ x) Λ (f Λ x)=c. Hence, S is distributive.

Theorem 3.3. Let L be a sub-semilattice of S. If J is an ideal of S/θl, I ={x ϵ S : πL (x) ϵ J} is an ideal of S. Then J is a proper ideal of S/θl if and only if I ∩ L=Ø.

Also, if P is a prime ideal of S such that P ∩ L=Ø then Q ={πL (x) ϵ S/θl: x ϵ P}is a prime ideal of S/θl.

Proof: Let J be an ideal of S/θl, and I ={x ϵ S : πL (x) ϵ J}, let a ϵ I, b ϵ S such that b ≤ a. Then b Λ x ≤ a for some

and hence b ϵ I. Therefore I is an initial segment of S. If πL (a) and πL (b) ϵ J;

Therefore J is an ideal. If a ϵ I ∩ L, then for any x ϵ S, πL (x) = πL (x Λ a) thus S/θl = J.

Conversely if S/θl = J, then for any x ϵ L, πL(x) ϵ J and therefore I ∩ L=Ø. Also, let P be a prime ideal of S such that P ∩ L=Ø, then Q is proper ideal of S/θl and let πL (a) Λ πL (b) ϵ Q, πL (a Λ b) ϵ Q.Then a Λ b ϵ P and since P is prime a ϵ P or b ϵ P.

Hence πL (a) ϵ Q or πL (b) ϵ Q. Therefore Q is prime.

Theorem 3.4. Let L and M be sub-semilattices of S such that L C M and let f: S/θl → S/θM if and only if I ∩ L=Ø. be the epimorphism defined by f (πL(x)) = πm(x),

If f is injection, then P∩L=Ø implies P∩M=Ø, for any prime ideal P of S.

Proof: Suppose P be a prime ideal S of such that P∩L=Ø then by Theorem 3.2 above, {πL (x) ϵ S/θl} is a prime ideal of S/θl and hence {πM (x) ϵ S/θM: x ϵ P} is a prime ideal of S/θM, so that P∩M=Ø.

Theorem: 3.5. Let F be a filter of S, then the following are equivalent:

1. F is a prime filter

2. F = {πF (x) : x ϵ S - F} is an ideal of S/θF

3. S/θF has a unique maximal ideal.

Proof: (i) -> (ii) Suppose that F is a prime filter, then S - F is a prime ideal.

Let πF (x) ϵ I and πF (y) ϵ S/θF such that πF (y) ≤ πF (x)

Therefore I is an initial segment. Let πF(x), πF(y) ϵI this implies x,y ϵ S - F. Then there exist z ϵ S - F such that x ≤ z and y ≤ z.

Again, there exists a ϵ F such that x Λ a = x Λ a Λ z and y Λ a = y Λ a Λ z

Therefore I = {πF (x) : x ϵ S - F} is an ideal of S/θF. Hence the result.

(ii) -> (iii): For any a ϵ F, πF (a) is the greatest element of S/θF. Hence, every ideal of S/θF is contained in {πF (x) : x ϵ S - P}.

Therefore S/θF has a unique maximal.

(iii)-> (i): Let G be the unique maximal ideal of S/θF and let x and y ϵ S - F. Since the principal ideals of S/θF generated by πF (x) and πF (y) are proper, and it follows that πF (x) and πF (y) ϵ G. Then, there exist πF (z) ϵ G such that πF (x) ≤ πF (z) and πF (y) ≤ πF (z). This implies x Λ z Λ a = x Λ a and y Λ z Λ a = y Λ a for some a ϵ F.

Then, there exist b ϵ S such that x≤ b, y≤ b and

, and thus S-F is a prime ideal. Henceforth F is prime filter.

Theorem 3.6. Let F be a filter of S. Then F is a maximal filter if and only if S/θF is the two-element chain.

Proof: Suppose F is a maximal filter, then for any x, and y ϵ F, πF (x) = πF (y) and for any x ϵ F and y ϵ S, πF (x) = πF (y) implies y ϵ F.

Now, let x and y ϵ S-F, Since F is maximal, there exist a,b ϵ F, such that a Λ x ≤ y and b Λ y ≤ x. Hence πF (x) = πF (a Λ x) ≤ πF (y) = πF (b Λ y) ≤ πF (x).

Therefore πF (x) = πF (y).

Conversely, suppose that S/θF is the two-element chain. Suppose x, and y ϵ S-F. Choose an element z ϵ F, such that πF (z) ≠ πF (x). Hence πF (x) = πF (y). Then there exist a ϵ F such that x Λ a = y Λ a and y ϵ F V [x).

Therefore, for any x ∉ F, F V [x) = S. Thus F is maximal. Hence the result.

Theorem 3.7. The map F↦θF is an isomorphism from the lattice F0(S) of all non-empty filters S of into a permutable sublattice of the lattice C(S) of all congruencies on S. Hence {θF : F ϵ F0 (S)}is a distributive and permutable sub lattice of C(S).

Proof: Suppose I and J are any filters of S. Then θi∩J C θi ∩ θj and let (x,y) ϵ θi ∩ θj, there exist a ϵ I, b ϵ J such that x Λ a = y Λ a and x Λ b = y Λ b.

Now since c ϵ I ∩ J, then (x,y) ϵ θi∩J. Hence F↦θF is a lattice homomorphism.

Also let us consider I, J ϵ F0(S) and θi C θj. Let x ϵ I, choose y ϵ J then there exist z ϵ I such that x≤ z and y≤ z.

Now (x,y) ϵ θi C θj and since z ϵ J, we have x ϵ J. There I C J fore and hence F↦θF is a lattice isomorphism of F0(S) onto the sub lattice {θF : F ϵ F0 (S)} of C(S).

Note that the lattice C(S) of all congruencies on a semilattice S need not be distributive, even when S is distributive. Hence the result.

Example 3.7. Let

, with partial order given by 0 < a < n, 0 < b < n, for all andwhen everis positive for

and m<n when ever m-n is positive for all

Then S becomes distributive semilattice, with aΛb=glb{a,b} Ɐ a,b ϵ S

CONCLUSION

In general, we introduced the notation of natural epimorphism of a semilattice on the quotient semilattice. Let L and M be sub-semilattices of S such that L C M and let f : S/θl → S/θm be the epimorphism deﬁned by f (πL (x)) = πM (x). Also, we can conclude that if f is injection map, then P∩L=Ø implies , P∩M=Ø for any prime ideal P of S. Additionally, for a distributive semilattice S, let F be a ﬁlter of S, then θF is the smallest congruence on S containing F in a single equivalence class and that S/θf is distributive. Similarly, map F↦θF is an isomorphism from the lattice F0(S) of all non-empty ﬁlters of into a permutable sublattice of S the lattice C(S) of all congruencies on S.

ACKNOWLEDGEMENT.

The authors would like to thank the editorial board and the anonymous referees for the comments and suggestions and constructive ideas on this paper. In particular, the correspondent author would like to express the special thanks to his wife, Ms. Tigist Tamirat for her support.

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