Lobachevskii Journal of Mathematics Vol. 13, 2003, 51 – 56

©Niovi Kehayopulu and Michael Tsingelis

ABSTRACT. An important problem for studying the structure of an ordered semigroup S is to know conditions under which for a given congruence ρ on S the set S∕ρ is an ordered semigroup. In [1] we introduced the concept of pseudoorder in ordered semigroups and we proved that each pseudoorder on an ordered semigroup S induces a congruence σ on S such that S∕σ is an ordered semigroup. In [3] we introduced the concept of semi-pseudoorder (also called pseudocongruence) in semigroups and we proved that each semi-pseudoorder on a semigroup S induces a congruence σ on S such that S∕σ is an ordered semigroup. In this note we prove that the converse of the last statement also holds. That is each congruence σ on a semigroup (S,.) such that S∕σ is an ordered semigroup induces a semi-pseudoorder on S.

For a given ordered semigroup (S,.,≤) is essential to know if there exists a congruence ρ on S such that S∕ρ be an ordered semigroup. This plays an important role for studying the structure of ordered semigroups. If S is a semigroup (resp. an ordered semigroup), by a congruence on S we mean an equivalence relation σ on S such that (a,b) ∈ σ implies (ac,bc) ∈ σ and (ca,cb) ∈ σ for all c ∈ S. If S is a semigroup and σ a congruence on S, then the set S∕σ := {(a)σ ∣a ∈ S} ( (a)σ is the σ-class of S containing a (a ∈ S)) is a semigroup and the operation on S∕σ is defined via the operation on S. The following question is natural: If (S,.,≤) is an ordered semigroup and σ a congruence on S, then is the set S∕σ an ordered semigroup ? A probable order on S∕σ could be the relation ” ≼ ” on S∕σ defined by means of the order ” ≤ ” on S, that is

But this relation is not an order, in general. An example can be found in [1]. The following question arises: Is there a congruence σ on S for which S∕σ is an ordered semigroup ? This let to the concept of pseudoorder introduced by the same authors in [1]. Let (S,.,≤) be an ordered semigroup. A relation ρ on S is called pseudoorder if

1. ≤⊆ ρ
2. (a,b) ∈ ρ and (b,c) ∈ ρ imply (a,c) ∈ ρ.
3. (a,b) ∈ ρ implies (ac,bc) ∈ ρ and (ca,cb) ∈ ρ for each c ∈ S.

According to Lemma 1 in [1], if (S,.,≤) is an ordered semigroup and σ a pseudoorder on S, then the relation σ̄ on S defined by

is a congruence on S and the set

S∕σ̄ is an ordered semigroup. So according to [1],

each pseudoorder on an ordered semigroup S induces a congruence σ̄ on S such that S∕σ̄ is an ordered semigroup. For a further study of pseudoorders in ordered semigroups we refer to [2]. On the other hand, the concept of pseudocongruences in semigroups has been introduced by the same authors in [3]. If (S,.) is a semigroup, by a pseudocongruence on S we mean a relation ρ on S such that

1. (a,a) ∈ ρ∀a ∈ S
2. (a,b) ∈ ρ and (b,c) ∈ ρ imply (a,c) ∈ ρ.
3. (a,b) ∈ ρ implies (ac,bc) ∈ ρ and (ca,cb) ∈ ρ for each c ∈ S.

If (S,.,≤) is an ordered semigroup, then each pseudoorder on

S is a pseudocongruence on S. Indeed, if ρ is a

pseudoorder on S and a ∈ S, then (a,a) ∈≤⊆ ρ. Pseudocongruences can be also called semi-pseudoorders, and from now on we will keep that terminology of semi-pseudoorders. We have seen in [3], that each semi-pseudoorder on a semigroup S induces a congruence ρ on S such that S∕ρ is an ordered semigroup. In this paper we prove that the converse of this statement also holds. For a semigroup (S,.) we define a multiplication ” ∗ ” on on S∕ρ defined by (a)ρ ∗ (b)ρ := (ab)ρ. If (S,.) is a semigroup and ρ a congruence on S and if there exists an order relation ”∇” on S∕ρ such that the (S∕ρ,∗,∇) is an ordered semigroup, then there exists a semi-pseudoorder σ on S such that ρ = σ̄. So each congruence ρ on a semigroup (S,.) such that S∕ρ is an ordered semigroup induces a semi-pseudoorder on S.

If (S,.) is a semigroup and σ a semi-pseudoorder on S, we define

The relation σ̄ is a congruence on S. Indeed: If a ∈ S, then (a,a) ∈ σ, then (a,a) ∈ σ−1, so (a,a) ∈ σ ∩ σ−1 := σ̄. If (a, b) ∈σ̄, then (a, b) ∈ σ and (a, b) ∈ σ−1, then (b, a) ∈ σ−1 and (b, a) ∈ σ, so (b, a) ∈ σ−1 ∩ σ := σ̄. If (a, b) ∈σ̄ and (b, c) ∈σ̄, then (a, b) ∈ σ, (a, b) ∈ σ−1, (b, c) ∈ σ, (b, c) ∈ σ−1, then (a, c) ∈ σ and (a, c) ∈ σ−1, thus (a, c) ∈ σ ∩ σ−1 := σ̄. Let (a, b) ∈σ̄ and c ∈ S. We have (a,b) ∈ σ and (a,b) ∈ σ−1. Since (a,b) ∈ σ, c ∈ S, we have (ac,bc) ∈ σ, (ca,cb) ∈ σ. Since (a, b) ∈ σ−1, we have (b,a) ∈ σ, then (bc,ac) ∈ σ, (ca,cb) ∈ σ, hence (ac,bc) ∈ σ−1, (ca,cb) ∈ σ−1. Then we have (ac,bc) ∈ σ ∩ σ−1 := σ̄ and (ca,cb) ∈ σ ∩ σ−1 := σ̄.

[It might be also noted that σ̄ = {(a,b) ∈ S × S ∣(a,b) ∈ σ and (b,a) ∈ σ}. Hence σ̄ is a congruence on S (cf. [3])]. Since σ̄ is a congruence on S, the set S∕σ̄ with the operation ” ∗ ” on S∕σ̄ defined by (a)σ̄ ∗ (b)σ̄ := (ab)σ̄ is a semigroup (It is known).

If (S,.) is a semigroup and σ a

semi-pseudoorder on S, we define a relation ”∇” on S∕σ̄ as follows: (a)σ̄∇(b)σ̄ if and only if (a,b) ∈ σ. The relation ”∇” on S∕σ̄ is well defined. Indeed: Let (a)σ̄ = (c)σ̄, (b)σ̄ = (d)σ̄ and (a)σ̄ ≼ (b)σ̄. Since (a)σ̄∇(b)σ̄, we have (a, b) ∈ σ. Since (a)σ̄ = (c)σ̄, we have (a, c) ∈σ̄ := σ ∩ σ−1 ⊆ σ−1, then (c, a) ∈ σ. Since (b)σ̄ = (d)σ̄, we have (b,d) ∈σ̄ := σ ∩ σ−1 ⊆ σ, then (b,d) ∈ σ. Then (c,d) ∈ σ, and (c)σ̄∇(d)σ̄. (Cf. also [3]).

Theorem. Let (S, .) be a semigroup. If σ is a semi-pseudoorder on S, then the set (S∕σ̄,∗,∇) is an ordered semigroup. Let ρ be a congruence on S and suppose there exists an order relation ” ≼ ” on S∕ρ such that (S∕ρ̄,∗,∇) be an ordered semigroup. Then there exists a semi-pseudoorder σ on S such that

Proof. For the first part of the Theorem we refer to the Theorem in [3]. Let now ρ be a congruence on S and ” ≼ ” an order on S∕ρ such that (S∕ρ,∗,≼) be an ordered semigroup. Let σ be the relation on S defined by

1) σ is a semi-pseudoorder on S. In fact:
Let a ∈ S. Since (a)ρ ≼ (a)ρ, we have (a,a) ∈ σ. Let (a,b) ∈ σ, (b, c) ∈ σ. Then (a)ρ ≼ (b)ρ, (b)ρ ≼ (c)ρ, then (a)ρ ≼ (c)ρ, and (a, c) ∈ σ. Let (a, b) ∈ σ and c ∈ S. Then (a)ρ ≼ (b)ρ and (c)ρ ∈ S∕ρ. Since (S∕ρ,∗,≼) is an ordered semigroup, we have (a)ρ ∗ (c)ρ ≼ (b)ρ ∗ (c)ρ, then (ac)ρ ≼ (bc)ρ, and (ac,bc) ∈ σ. Similarly (a,b) ∈ σ and c ∈ S, imply (ca,cb) ∈ σ.

2) ρ = σ̄. Indeed: We have

3) ≼ = ∇. Indeed:
Let (a)ρ ≼ (b)ρ. Since (a,b) ∈ σ, we have (a)σ̄∇(b)σ̄. By 2), ρ = σ̄. So (a)σ̄ = (a)ρ and (b)σ̄ = (b)ρ. Then (a)ρ∇(b)ρ.
Let (a)ρ∇(b)ρ. Since ρ = σ̄, we have (a)ρ = (a)σ̄ and (b)ρ = (b)σ̄. Then (a)σ̄∇(b)σ̄, hence (a,b) ∈ σ, and (a)ρ ≼ (b)ρ.

Remark 1. If (S,.,≤) is an ordered semigroup and ρ a pseudoorder on S, then the mapping

is a homomorphism. In fact, if a,b ∈ S, then

Let now a ≤ b. Since (a,b) ∈≤⊆ ρ, we have (a,b) ∈ ρ. Then, since ρ is a semipseudoorder on S, we have (a)ρ̄∇f(b), and f(a)∇f(b). □

For a semigroup S, we denote by SP(S) the set of semi-pseudoorders on S and by C(S) the set of congruences on S. Let ” ≈ ” be the equivalence relation on S defined as follows:

Remark 2. If S is a semigroup and ρ a semi-pseudoorder on S, then the mapping

is (1-1) and onto. In fact: The mapping f is well defined: If ρ is a semi-pseudoorder on S, then ρ̄ is a congruence on S. Let ρ,σ ∈SP(S) and (ρ)≈ = (σ)≈. Then we have ρ ≈ σ, and ρ̄ = σ̄.
f is (1-1): Let ρ,σ ∈SP(S) such that ρ̄ = σ̄. Then ρ ≈ σ, and (ρ)≈ = (σ)≈.
f is onto: Let ρ ∈C(S). Then ρ = ρ−1 and ρ is a semi-pseudoorder on S. Thus ρ ∈SP(S) and

□

For a semigroup S, we denote by OC(S) the set of all congruences ρ on S for which there exists an order relation ”∇” on S∕ρ such that (S∕ρ,∗,∇) is an ordered semigroup.

Remark 3. If S is a semigroup, then the mapping

is (1-1) and onto. In fact: The mapping f is well defined: If ρ is a semi-pseudoorder on S, then ρ ̄ is a congruence on S. Then, by the Theorem, the set (S∕ρ̄,∗,∇) is an ordered semigroup. Which means that ρ̄ ∈OC(S).
Let ρ,σ ∈SP(S) and (ρ)≈ = (σ)≈. Then we have ρ ≈ σ, and ρ̄ = σ̄.
f is (1-1): Let ρ,σ ∈SP(S) and ρ̄ = σ̄. Then ρ ≈ σ, and (ρ)≈ = (σ)≈.
f is onto: Let ρ ∈OC(S). By the Theorem, there exists a semi-pseudoorder σ on S such that ρ = σ̄. Then σ ∈SP(S), and f((σ))≈ := σ̄ = ρ.

Acknowledgment. This research was supported by the Special Research Account of the University of Athens (Grant No. 70/4/5630).

References

UNIV. OF ATHENS, DEPT. OF MATHEMATICS HOME: NIOVI KEHAYOPULU, NIKOMIDIAS 18, 161 22 KESARIANI, GREECE

E-mail address: nkehayop@cc.uoa.gr

ReceivedSeptember 30, 2003