PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97 (111) (2015), 199–210 DOI: 10.2298/PIM140421001S

ON STURDY FRAME OF ABSTRACT ALGEBRAS

Yong Shao and Miaomiao Ren

Abstract. We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an application, we study the sturdy distributive lattice of lattice ordered groups.

1. Introduction and preliminaries

The union of algebras of the same type has been studied by many algebraists. The sturdy semilattice of semigroups is introduced by Petrich in [14]. It is an important tool to study the structures of semigroups, for example, see [15]. Pastijn [12] introduces the sum of a lattice ordered system. Ghosh [3] and Guo, Sen, and Shum [19] introduce the concept of strong distributive lattice of semirings, respectively. By using this concept, Guo, Sen, and Shum [5, 19] study structures of idempotent semirings. Zhao, Guo, and Shum [22] introduce and study sturdy frame of type (2, 2) algebras. By introducing strong b-lattice of semirings, Sen, Maity, and Shum [20] study generalized Clifford semirings. We introduce and study a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings.

Throughout this paper, unless otherwise stated, we consider abstract algebras and terms of a fixed type F without nullary operation symbols. For an algebra A, we shall denote the universe of A by A. Moreover, we shall write the symbols of mappings on the right and the symbols of operations on the left.

2010 Mathematics Subject Classification: 03C05; 08A05.

Key words and phrases: abstract algebra, sturdy frame, congruence, subdirect product, variety.

The first author is supported by China Postdoctoral Science Foundation, Grant 2011M501466 and the Natural Science Foundation of Shannxi Province, Grant 2011JQ1017.

Communicated by Siniša Crvenković.

An algebra A is called idempotent if f^{A} (a, …, a) = a for any n-ary f ∈ F and any a ∈ A. Bands, idempotent semirings and lattices are examples of idempotent algebras. By a frame B we mean an idempotent algebra endowed with an upper semilattice order ⩽ satisfying f^{B}(b_{1}, …, b_{n}) ⩽ b_{1} ∨ ⋯∨ b_{n} for any n-ary f ∈ F and any b_{1}, …, b_{n} ∈ B, where b_{1} ∨ ⋯∨ b_{n} = lub{b_{1}, …, b_{n}}. It is easy to see that semilattices and lattices are frames.

Let B be a frame and {A_{α} | α ∈ B} a family of pairwise disjoint algebras, indexed by B. For each pair α, β of elements of B such that α ⩽ β, let φ_{α,β} : A_{α} → A_{β} be a monomorphism, and assume that

(a) φ_{α,α} = 1_{A_{α}} for every α ∈ B;

(b) φ_{α,β} φ_{β,γ} = φ_{α,γ} for every α, β, γ ∈ B such that α ⩽ β ⩽ γ;
(c) If n-ary f ∈ F and α_{1} ∨ ⋯∨ α_{n} ⩽ γ for α_{1}, …, α_{n}, γ ∈ B, then
f^{A_{γ}} (a_{1}φ_{α_{1} ,γ} , …, a_{n}φ_{α_{n} ,γ} ) ∈ (A_{f^{B} (α_{1} ,...,α_{n} )} )φ_{f^{B} (α_{1} ,...,α_{n} ),γ}
for any a_{i} ∈ A_{α_{i}} , 1 ⩽ i ⩽ n.

Let A = ⋃ _{α∈B} A_{α}, and define an n-ary operation f on A by

f^{A} (a_{1}, …, a_{n}) = f^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{f^{B} (α_{1} ,...,α_{n} ),α}^{−1}

for any a_{i} ∈ A_{α_{i}} , 1 ⩽ i ⩽ n, where α = α_{1} ∨ ⋯∨ α_{n}. Then we can check that A = ⟨A, F ⟩ is an algebra of type F, denoted by A = [B, ⩽; A_{α} , φ_{α,β} ]. We call the constructed algebra A = [B, ⩽; A_{α} , φ_{α,β} ] the sturdy frame of algebras A_{α}.

It is easy to see that the sturdy semilattice of semigroups introduced by Petrich [14], the sum of lattice ordered systems introduced by Pastijn [12], the strong distributive lattice of semirings introduced by Ghosh [3] and Guo, Sen and Shum [19], the strong b-lattice of semirings introduced by Sen, Maity and Shum [20] and the sturdy frames of type (2, 2) algebras introduced by Zhao, Guo and Shum [22] are all special cases of the sturdy frame of algebras. Thus the sturdy frame of algebras provides a new tool to investigate the structures of algebras.

If A = [B, ⩽; A_{α} , φ_{α,β} ], then it is easy to see that every A_{α} is a subalgebra of A. Also, suppose that the frame B satisfies the additional condition

f^{B}(α_{1}, …α_{n}) = α_{1} ∨ ⋯∨ α_{n}

for any n-ary f ∈ F and any α_{1}, …, α_{n} ∈ B. Then the algebra [B, ⩽; A_{α} , φ_{α,β} ] coincides with the Płonka sum of the direct systems ⟨B, ⩽; A_{α} , φ_{α,β} ⟩ [17].

For notations and terminologies not given in this paper, the reader is referred to Burris and Sankappanavar [2] and Grätzer [6] for information concerning universal algebra, to Howie [8] and Petrich [15] for a background on semigroup theory and to Hebisch and Weinert [7] for knowledge on semiring theory, respectively. We shall assume that the reader is familiar with the basic results in these areas.

2. Properties and characterizations of sturdy frame of algebras

In this section we give some properties and characterizations of sturdy frame of algebras.

Proposition 2.1. Let A = [B, ⩽; A_{α} , φ_{α,β} ] and t an n-ary term. Then

t^{A} (a_{1}, …, a_{n}) = t^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t^{B} (α_{1} ,...,α_{n} ),α}^{−1}

for any a_{i} ∈ A_{α_{i}} (1 ⩽ i ⩽ n), where α = α_{1} ∨ ⋯∨ α_{n}.

Proof. We prove it by induction on l(t) (the length of t). If l(t) = 0, then t = x_{i} for some i. Further, we have that t^{A} (a_{1}, …, a_{n}) = a_{i} and that

t^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t^{B} (α_{1} ,...,α_{n} ),α}^{−1} = a_{i}φ_{α_{i} ,α} φ_{α_{i} ,α}^{−1} = a_{i}.

It follows that t^{A} (a_{1}, …, a_{n}) = t^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t^{B} (α_{1} ,...,α_{n} ),α}^{−1} .

Suppose that l(t) ⩾ 1 and that the result holds for every term w with l(w) < l(t) . Then t is the form of t = f(t_{1}(x_{1}, …, x_{n}), …, t_{k}(x_{1}, …, x_{n})), where f is an k-ary operation symbol in F. Since l(t_{i}) < l(t), we must have that for any i (1 ⩽ i ⩽ k),

t_{i}^{A} (a_{1}, …, a_{n}) = t_{i}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{i}^{B} (α_{1} ,...,α_{n} ),α}^{−1} .

Put β = t_{1}^{B}(α_{1}, …, α_{n}) ∨ ⋯∨ t_{k}^{B}(α_{1}, …, α_{n}). We have

t^{A}(a_{1}, …, a_{n})φ_{t^{B} (α_{1} ,...,α_{n} ),α}

= f^{A} (t_{1}^{A} (a_{1}, …, a_{n}), …, t_{k}^{A} (a_{1}, …, a_{n}))φ_{t^{B} (α_{1} ,...,α_{n} ),α}

= f^{A} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α}^{−1} , …,

t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α}^{−1} )φ_{t^{B} (α_{1} ,...,α_{n} ),α}

= f^{A_{β}} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),β} , …,

t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),β} )
φ_{t^{B} (α_{1} ,...,α_{n} ),β}^{−1} φ_{t^{B} (α_{1} ,...,α_{n} ),α}

= f^{A_{β}} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),β} , …,

t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),β} )
φ_{t^{B} (α_{1} ,...,α_{n} ),β}^{−1} φ_{t^{B} (α_{1} ,...,α_{n} ),β} φ_{β,α}

= f^{A_{β}} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{1}^{B} (α_{1} ,...,α_{n} ,β)} , …,

t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{k}^{B} (α_{1} ,...,α_{n} ,β)} )φ_{β,α}

= f^{A_{α}} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{1}^{B} (α_{1} ,...,α_{n} ,β)} φ_{β,α} , …,

t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{k}^{B} (α_{1} ,...,α_{n} ,β)} φ_{β,α} )

= f^{A_{α}} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{1}^{B} (α_{1} ,...,α_{n} ),α} , …,

t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{t_{k}^{B} (α_{1} ,...,α_{n} ),α} )

= f^{A_{α}} (t_{1}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} ), …, t_{k}^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} ))

= t^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} ).

Consequently, t^{A} (a_{1}, …, a_{n}) = t^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{t^{B} (α_{1} ,...,α_{n} ),α}^{−1} . □

In the following we show that the sturdy frame of algebras can be represented as a subdirect product of two algebras.

Theorem 2.1. Let A = [B, ⩽; A_{α} , φ_{α,β} ], a ∈ A_{α} and b ∈ A_{β}. Define binary relations ρ and θ on A by

(a, b) ∈ ρ ⇔ α = β,
(a, b) ∈ θ ⇔ aφ_{α,α∨β} = bφ_{β,α∨β} .

Then ρ and θ are congruences on A and A is a subdirect product of B and A/θ. If each A_{α} satisfies an identity p ≈ q, so does A/θ.

Proof. It is easy to verify that ρ is a congruence on A and that A/ρ is isomorphic to B. Also, it is clear that θ is reflexive and symmetric. To show that θ is transitive, let a ∈ A_{α} , b ∈ A_{β} and c ∈ A_{γ} such that (a, b) ∈ θ and (b, c) ∈ θ. Then aφ_{α,α∨β} = bφ_{β,α∨β} and bφ_{β,β∨γ} = cφ_{γ,β∨γ} . Further, we have

aφ_{α,α∨β∨γ} = aφ_{α,α∨β} φ_{α∨β,α∨β∨γ} = bφ_{β,α∨β} φ_{α∨β,α∨β∨γ} = bφ_{β,α∨β∨γ}
= bφ_{β,β∨γ} φ_{β∨γ,α∨β∨γ} = cφ_{γ,β∨γ} φ_{β∨γ,α∨β∨γ} = cφ_{γ,α∨β∨γ} .

It follows that

aφ_{α,α∨γ} = aφ_{α,α∨β∨γ} φ_{α∨γ,α∨β∨γ}^{−1} = cφ_{γ,α∨β∨γ} φ_{α∨γ,α∨β∨γ}^{−1} = cφ_{γ,α∨γ}

and so (a, c) ∈ θ. This shows that θ is transitive. Thus θ is an equivalence on A.

Let a_{i} ∈ A_{α_{i}} and b_{i} ∈ A_{β_{i}} such that (a_{i}, b_{i}) ∈ θ, 1 ⩽ i ⩽ n. Then a_{i}φ_{α_{i} ,α_{i} ∨β_{i}} = b_{i}φ_{β_{i} ,α_{i} ∨β_{i}} . Put α = α_{1} ∨ ⋯∨ α_{n} and β = β_{1} ∨ ⋯∨ β_{n}. We have

a_{i}φ_{α_{i} ,α_{i} ∨β_{i}} φ_{α_{i} ∨β_{i} ,α∨β} = b_{i}φ_{β_{i} ,α_{i} ∨β_{i}} φ_{α_{i} ∨β_{i} ,α∨β} .

This implies that a_{i}φ_{α_{i} ,α∨β} = b_{i}φ_{β_{i} ,α∨β} .

Choose γ = f^{B}(α_{1}, …, α_{n}) ∨ f^{B}(β_{1}, …, β_{n}). We have

f^{A} (a_{1}, …, a_{n})φ_{f^{B} (α_{1} ,...,α_{n} ),γ}
= f^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{f^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{f^{B} (α_{1} ,...,α_{n} ),γ}
= f^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{f^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{f^{B} (α_{1} ,...,α_{n} ),α∨β} φ_{γ,α∨β}^{−1}
= f^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{f^{B} (α_{1} ,...,α_{n} ),α}^{−1} φ_{f^{B} (α_{1} ,...,α_{n} ),α} φ_{α,α∨β} φ_{γ,α∨β}^{−1}
= f^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )φ_{α,α∨β} φ_{γ,α∨β}^{−1}
= f^{A_{α∨β}} (a_{1}φ_{α_{1} ,α} φ_{α,α∨β} , …, a_{n}φ_{α_{n} ,α} φ_{α,α∨β} )φ_{γ,α∨β}^{−1}
= f^{A_{α∨β}} (a_{1}φ_{α_{1} ,α∨β} , …, a_{n}φ_{α_{n} ,α∨β} )φ_{γ,α∨β}^{−1}
= f^{A_{α∨β}} (b_{1}φ_{β_{1} ,α∨β} , …, b_{n}φ_{β_{n} ,α∨β} )φ_{γ,α∨β}^{−1}
= f^{A_{α∨β}} (b_{1}φ_{β_{1} ,β} φ_{β,α∨β} , …, b_{n}φ_{β_{n} ,β} φ_{β,α∨β} )φ_{γ,α∨β}^{−1}
= f^{A_{β}} (b_{1}φ_{β_{1} ,β} , …, b_{n}φ_{β_{n} ,β} )φ_{β,α∨β} φ_{γ,α∨β}^{−1}
= (f^{A_{β}} (b_{1}φ_{β_{1} ,β} , …, b_{n}φ_{β_{n} ,β} )φ_{f^{B} (β_{1} ,...,β_{n} ),β}^{−1} )φ_{f^{B} (β_{1} ,...,β_{n} ),β} φ_{β,α∨β} φ_{γ,α∨β}^{−1}

= f^{A} (b_{1}, …, b_{n})φ_{f^{B} (β_{1} ,...,β_{n} ),β} φ_{β,α∨β} φ_{γ,α∨β}^{−1}
= f^{A} (b_{1}, …, b_{n})φ_{f^{B} (β_{1} ,...,β_{n} ),α∨β} φ_{γ,α∨β}^{−1} = f^{A} (b_{1}, …, b_{n})φ_{f^{B} (β_{1} ,...,β_{n} ),γ} .

Thus (f^{A} (a_{1}, …, a_{n}), f^{A} (b_{1}, …, b_{n})) ∈ θ and so θ is a congruence on A. Notice that ρ ∩ θ = Δ, where Δ is the equality relation. Hence A is a subdirect product of B and A/θ.

Let p(x_{1}, …, x_{n}) ≈ q(x_{1}, …, x_{n}) be an identity. In the following we shall show that A satisfies p ≈ q if each A_{α} satisfies p ≈ q. In fact, we have

p^{A/θ} (a_{1}/θ, …, a_{n}/θ) = p^{A/θ} (a_{1}φ_{α_{1} ,α} /θ, …, a_{n}φ_{α_{n} ,α} /θ)

= p^{A} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )/θ = p^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )/θ.

Similarly, q^{A/θ} (a_{1}/θ, …, a_{n}/θ) = q^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} )/θ. Since A_{α} satisfies p ≈ q, it follows that p^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} ) = q^{A_{α}} (a_{1}φ_{α_{1} ,α} , …, a_{n}φ_{α_{n} ,α} ). Thus

p^{A/θ} (a_{1}/θ, …, a_{n}/θ) = q^{A/θ} (a_{1}/θ, …, a_{n}/θ)

and so A/θ satisfies p ≈ q. □

Theorem 2.1 generalizes and enriches Lemma I.8.11 in [14], Lemma 2.6 in [20] and Theorem 2.2 in [22], respectively. If A = [B, ⩽; A_{α} , φ_{α,β} ], then it is easy to verify that A/θ in which θ is defined in Theorem 2.1 is a direct limit of the family {A_{α} | α ∈ B} [6]. By Theorem 2.1, we can immediately have the following result, which generalizes and enriches Theorem 1.2 in [3], Lemma 3.2 in [19], Theorem 2.4 in [20] and Theorem 2.3 in [22], respectively.

Corollary 2.1. Let A = [B, ⩽; A_{α} , φ_{α,β} ] and p ≈ q an identity. Then the following statements are equivalent:

(i) B and each algebra A_{α} satisfy p ≈ q;

(ii) B and A/θ satisfy p ≈ q;

(iii) A satisfies p ≈ q.

A variety is said to be a frame variety if every member in it can become a frame under some upper semilattice order. The variety of semilattices and the variety of lattices are examples of a frame variety. Every member in a variety V will be called a V -algebra. For a variety V and an algebra A there exists the smallest congruence ρ on A such that A/ρ is a V -algebra. This congruence will be called the least V - congruence on A. The following theorem characterizes a sturdy frame of algebras by subdirect product decomposition.

Theorem 2.2. Let A be an algebra, let V be a variety and W a frame variety. Assume that τ_{1} is the least V -congruence on A and that τ_{2} is the least W -congruence on A. Then the following statements are equivalent:

(i) A is the subdirect product of A/τ_{1} and A/τ_{2}; (ii) A can be expressed as the sturdy frame [B, ⩽; A_{α} , φ_{α,β} ] of V -algebras A_{α}

on frame B in W; (iii) A is a subdirect product of a V -algebra and a W -algebra.

Proof. (i) ⇒ (ii). By hypothesis, it follows that A/τ_{2} can become a frame under the upper semilattice order ⩽. We shall denote A/τ_{2} by B and A/τ_{1} by C, respectively. For any α ∈ B, let A_{α} denote the algebra whose universe is {α} × C ∩ A. It is easy to see that A_{α} belongs to V. Also, it is routine to verify that A is the union of all A_{α}, every A_{α} is a subalgebra of A and that A_{α} ∩ A_{β} = ∅ if α ≠ β. Let α, β, γ ∈ B such that α ⩽ β ⩽ γ. Define a mapping φ_{α,β} : A_{α} → A_{β} by (α, c)φ_{α,β} = (β, c) ((α, c) ∈ A_{α}). It is clear that φ_{α,β} is injective and that φ_{α,β} φ_{β,γ} = φ_{α,γ} . Moreover, for any (α, c_{i}) ∈ A_{α} (1 ⩽ i ⩽ n), we have

f^{A_{α}} ((α, c_{1}), …, (α, c_{n}))φ_{α,β} = (f^{B}(α, …, α), f^{C}(c_{1}, …, c_{n}))φ_{α,β}

= (α, f^{C}(c_{1}, …, c_{n}))φ_{α,β} = (β, f^{C}(c_{1}, …, c_{n}))
= (f^{B}(β, …, β), f^{C}(c_{1}, …, c_{n})) = f^{A_{β}} ((β, c_{1}), …, (β, c_{n}))
= f^{A_{β}} ((α, c_{1})φ_{α,β} , …, (α, c_{n})φ_{α,β} ).

This shows that φ_{α,β} is a monomorphism.

If α_{1} ∨ ⋯∨ α_{n} ⩽ γ, then

f^{A_{γ}} ((α_{1}, c_{1})φ_{α_{1} ,γ} , …, (α_{n}, c_{n})φ_{α_{n} ,γ} ) = f^{A_{γ}} ((γ, c_{1}), …, (γ, c_{n}))
= (f^{B}(γ, …, γ), f^{C}(c_{1}, …, c_{n})) = (γ, f^{C}(c_{1}, …, c_{n}))
= (f^{B}(α_{1}, …, α_{n}), f^{C}(c_{1}, …, c_{n}))φ_{f^{B} (α_{1} ,...,α_{n} ),γ}
∈ (A_{f^{B} (α_{1} ,...,α_{n} )} )φ_{f^{B} (α_{1} ,...,α_{n} ),γ} .

Put α = α_{1} ∨ ⋯∨ α_{n}. We have

f^{A} ((α_{1}, c_{1}), …, (α_{n}, c_{n}))φ_{f^{B} (α_{1} ,...,α_{n} ),α}
= f^{B×C} ((α_{1}, c_{1}), …, (α_{n}, c_{n}))φ_{f^{B} (α_{1} ,...,α_{n} ),α}
= (f^{B}(α_{1}, …, α_{n}), f^{C}(c_{1}, …, c_{n}))φ_{f^{B} (α_{1} ,...,α_{n} ),α}
= (α, f^{C}(c_{1}, …, c_{n})) = (f^{B}(α, …, α), f^{C}(c_{1}, …, c_{n}))
= f^{A_{α}} ((α, c_{1}), …, (α, c_{n})) = f^{A_{α}} ((α_{1}, c_{1})φ_{α_{1} ,α} , …, (α_{n}, c_{n})φ_{α_{n} ,α} ).

Thus A can be expressed as the sturdy frame [B, ⩽; A_{α} , φ_{α,β} ] of algebras A_{α} in V on frame B in W.

(ii) ⇒ (iii). This follows from Theorem 2.1 immediately.

(iii) ⇒ (i). Assume that A is a subdirect product of a V -algebra and a W - algebra. Then there exist congruences τ̃_{1} and τ̃_{2} on A such that A/τ̃_{1} ∈ V, A/τ̃_{2} ∈ W and τ̃_{1} ∩ τ̃_{2} = Δ. Since τ_{1} ⊆ τ̃_{1} and τ_{2} ⊆ τ̃_{2}, it follows that τ_{1} ∩ τ_{2} = Δ. Thus A is the subdirect product of A/τ_{1} and B/τ_{2}. □

As a corollary, we have the following corresponding result for semigroups, which generalizes some results obtained by Petrich and Reilly [16].

Corollary 2.2. Let S be a semigroup and V a semigroup variety. Then S is a sturdy semilattice of semigroups in V if and only if S is a subdirect product of a semilattice and a semigroup in V.

By a semiring we mean an algebra (S, +, ·) with two binary operations + and · such that both the additive reduct (S, +) and the multiplicative reduct (S, ·) are semigroups and such that the following distributive laws hold:

x(y + z) ≈ xy + xz, (y + z)x ≈ yx + zx.

A semiring S is said to be idempotent if both (S, +) and (S, ·) are bands. The class of all idempotent semirings whose additive reducts are semilattices will be denoted by Sl^{+} . Let S ∈ Sl^{+} . Define an upper semilattice order ⩽ on S by

(2.1)  a ⩽ b ⇔ a + b = b.

By Lemma 3.3 in [22] we have that (S, ⩽) is a frame. The class of all idempotent semirings for which the two reducts are semilattices will be denoted by Bi. Given S ∈ Bi. Define an upper semilattice order ⩽ on S by

(2.2)  a ⩽ b ⇔ ab = b.

It follows from Lemma 4.2 in [22] that (S, ⩽) is a frame. By Theorem 2.2 we can immediately obtain the following result for semirings.

Corollary 2.3. Let S be a semiring and V a semiring variety. Then S is a sturdy frame of semirings in V on a frame which is described by (2.1)/ (2.2) if and only if S is a subdirect product of a semiring in Sl^{+} [Bi] and a semiring in V.

It is clear that Sl^{+} coincides with the class of all b-lattices, which are introduced in [20] and that the variety of all distributive lattices is a subvariety of Bi. Consequently, Corollary 2.3 extends and enriches some results obtained by Bandelt and Petrich [1], Ghosh [3], Guo, Sen, and Shum [5, 19], Sen, Maity, and Shum [20] and Shao and Zhao [21], respectively.

3. Sturdy distributive lattice of lattice ordered groups

As an application of sturdy frame of algebras, the sturdy distributive lattice of lattice ordered groups will be investigated in this section.

Recall [8, 9] that a partially ordered semigroup S is said to be a ∨-semilatticed semigroup if there exists the least upper bound a ∨ b for each pair of elements a, b ∈ S and if the multiplication distributes over the join operation ∨, that is,

(∀a, b, c ∈ S) a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc.

In a dual way, we may consider ∧-semilatticed semigroups. A ∨-semilatticed semigroup or a ∧-semilatticed semigroup is simply called a semilatticed semigroup. In particular, if a partially ordered semigroup S is both a ∨-semilatticed semigroup and a ∧-semilatticed semigroups, then S is called an lattice ordered semigroup. We denote by (S, ∨, ∧, ·) the lattice ordered semigroup S.

Suppose that (S, ·) is an inverse semigroup. We denote by ⪯ the natural partial order on S. That is to say (Section II.4 in [15]),

a ⪯ b ⇔ (∃e, f ∈ E) a = be = fb

holds for any a, b ∈ S, in which E is the set of idempotents of (S, ·). It is easy to verify that (∀a, b, c ∈ S) a ⪯ b ⇒ ac ⪯ bc, ca ⪯ cb. Suppose that (S, ∨, ·) is a ∨-semilatticed semigroup. We denote by L (R, D and H, respectively) denotes Green’s L-relation (R-relation, D-relation, H-relation, respectively) on the multiplicative reduct (S, ·) of S.

Suppose that (B, ∨, ·) is a ∨-semilatticed semilattice. If B satisfies the absorption law x ∨ xy ≈ x, then (B, ∨, ·) is called a distributive lattice.

Suppose that (S, ∨, ·) is a ∨-semilatticed inverse semigroup under the partial order ⩽. Let E(S) be the set of idempotents of the multiplicative reduct (S, ·) of S, i.e., E(S) = {e ∈ S | e^{2} = e}.

Thus we have directly the following result from Theorem 2.3 in [3].

Theorem 3.1. If (S, ∨, ·) is a ∨-semilatticed inverse semigroup under the partial order ⩽, then (E(S), ∨, ·) is a semilatticed semilattice.

When E(S) is a distributive lattice for a ∨-semilatticed inverse semigroup S, we have

Proposition 3.1. If (S, ∨, ·) is a ∨-semilatticed inverse semigroup under the partial order ⩽, then the following conditions are equivalent:

(i) ⩽ is an extension of ⪯ (i.e., ⪯ ⊆ ⩽); (ii) (∀e, f ∈ E(S)) e ⩽ f ⇔ e ⪯ f; (iii) (∀a, b ∈ S) a ∨ ab^{−1} b = a; (iv) (E(S), ∨, ·) is a distributive lattice.

Proof. Suppose that (S, ∨, ·) is a ∨-semilatticed inverse semigroup under the partial order ⩽.

(i) ⇒ (ii). Suppose that e, f ∈ E(S). If e ⪯ f, then it follows immediately from (i) that e ⩽ f. Conversely, if e ⩽ f, then pre-multiplying this by e, we have e ⩽ ef. On the other hand, since ef ⪯ e, it follows from (i) that ef ⩽ e. Thus, ef = e holds. That is to say, e ⪯ f. This shows that e ⩽ f ⇔ e ⪯ f, as required.

(ii) ⇒ (i). Suppose that a, b ∈ S. If a ⪯ b, then it follows from Proposition 5.2.1 in [8] that b^{−1} a = a^{−1} a, aa^{−1} ⪯ bb^{−1} and a^{−1} a ⪯ b^{−1} b. Thus we have immediately from (ii) that a^{−1} a ⩽ b^{−1} b, aa^{−1} ⩽ bb^{−1} . Hence,

a ⩽ bb^{−1} a (post-multiplying aa^{−1} ⩽ bb^{−1} by a)
⩽ ba^{−1} a (b^{−1} a = a^{−1} a)
⩽ bb^{−1} b (a^{−1} a ⩽ b^{−1} b)
= b.

That is to say, a ⩽ b. This shows that ⩽ is an extension of ⪯.

(ii) ⇒ (iii). It is clear that a^{−1} ab^{−1} b ⪯ a^{−1} a for any a, b ∈ S. Thus it follows directly from (ii) that a^{−1} ab^{−1} b ⩽ a^{−1} a. That is to say, a^{−1} ab^{−1} b ∨ a^{−1} a = a^{−1} a. Premultiplying this by a, we have a(a^{−1} ab^{−1} b ∨ a^{−1} a) = ab^{−1} b ∨ a = a(a^{−1} a) = a. Hence, ab^{−1} b ∨ a = a, as required.

(iii) ⇒ (iv). It is clear from Theorem 3.1 that (E(S), ∨, ·) is a semilatticed semilattice. Suppose that e, f ∈ E(S). Then it follows directly from (iii) that ef ∨ e = e since f^{−1} = f. That is to say, E(S) satisfies the absorption law and so it is a distributive lattice, as required.

(iv) ⇒ (ii). Suppose that E(S) is a distributive lattice and e, f ∈ E(S). If e ⩽ f, then e ∨ f = f. Pre-multiplying this by e, we can show e ∨ ef = ef. Since the absorption law is satisfied in E(S), it follows that e = e∨ef = ef and so e ⪯ f. Conversely, if e ⪯ f, then ef = e. This implies that ef ∨ f = e ∨ f. By using absorption law again, we have f = ef ∨ f = e ∨ f. That is to say, e ⩽ f. This shows that e ⩽ f ⇔ e ⪯ f, as required. □

McAlister introduced amenable partial orders on inverse semigroups and studied amenable partially ordered inverse semigroups in [10], in which, amenable partial order is an extension of the natural partial order. McAlister gave the definition of amenable partial order on an inverse semigroup as follows.

Definition 3.1. Let (S, ·, ⩽) be a partially ordered inverse semigroup. The partial order ⩽ is said to be a left(right) amenable partial order if it coincides with ⪯ on idempotents and for each a, b ∈ S, a ⩽ b implies a^{−1} a ⪯ b^{−1} b (aa^{−1} ⪯ bb^{−1} ). If ⩽ is both a left amenable partial order and a right amenable partial order on S, then ⩽ is called an amenable partial order and S is called an amenable partially ordered inverse semigroup.

Suppose that S is a Clifford semigroup. It is easy to see that both the left amenable partial order and the right amenable partial order coincide since Clifford semigroup satisfies aa^{−1} = a^{−1} a for any a ∈ S. Thus we have

Lemma 3.1. Suppose that (S, ∨, ·) a ∨-semilatticed Clifford semigroup under the amenable partial order ⩽. Then (E(S), ∨, ·) is a distributive lattice and S satisfies

(∀a, b ∈ S) (a ∨ b)^{−1} (a ∨ b) = a^{−1} a ∨ b^{−1} b.

Proof. Since ⩽ is amenable, it follows from Definition that ⩽ coincides with ⪯ on idempotents. By Propositon 3.1, we have that E(S) is a distributive lattice.

Suppose that a, b ∈ S. It is clear that a, b ⩽ a ∨ b. Since ⩽ is left amenable, it follows that a^{−1} a, b^{−1} b ⪯ (a ∨ b)^{−1} (a ∨ b). Thus we have that a^{−1} a, b^{−1} b ⩽ (a ∨ b)^{−1} (a ∨ b) and so a^{−1} a ∨ b^{−1} b ⩽ (a ∨ b)^{−1} (a ∨ b). On the other hand, it is obvious that ab^{−1} b ⪯ a, ba^{−1} a ⪯ b. It follows that ab^{−1} b ⩽ a, ba^{−1} a ⩽ b, since ⩽ extends the natural partial order ⪯. Thus we have that (a ∨ b)(a^{−1} a ∨ b^{−1} b) = a ∨ ab^{−1} b ∨ ba^{−1} a ∨ b = a ∨ b. This implies that (a ∨ b)^{−1} (a ∨ b)(a^{−1} a ∨ b^{−1} b) = (a∨b)^{−1} (a∨b) and so (a∨b)^{−1} (a∨b) ⪯ a^{−1} a∨b^{−1} b. By Proposition 3.1, we have that (a ∨ b)^{−1} (a ∨ b) ⩽ a^{−1} a ∨ b^{−1} b. This shows that (a ∨ b)^{−1} (a ∨ b) = a^{−1} a ∨ b^{−1} b. □

Suppose that (S, ∨, ·) a ∨-semilatticed Clifford semigroup under the amenable partial order ⩽. Since the multiplicative reduct of S is a Clifford semigroup, it follows that aa^{−1} = a^{−1} a for any a ∈ S. By Theorem II.1.4 in [15], we have that H is the least semilattice congruence of the Clifford semigroup (S, ·) and every H-class is a maximal subgroup of (S, ·). For any a ∈ S, H_{a} denotes the H-class containing a, and a^{0} denotes the identity of subgroup H_{a}. It can be easily seen that a H b if and only if a^{0} = b^{0} for any a, b ∈ S. Thus we have that E(S) = {a^{0} | a ∈ S}. By Lemma 3.1 we have

Corollary 3.1. Suppose that (S, ∨, ·) a ∨-semilatticed Clifford semigroup under the partial order ⩽. If ⩽ is amenable then (∀a, b ∈ S) (a ∨ b)^{0} = a^{0} ∨ b^{0}.

By [4], we have

Lemma 3.2. Suppose that (G, ·, ⩽) is a partially ordered group and a, b ∈ G. Then the following statements are equivalent:

(i) there exists the least upper bound a ∨ b of a and b; (ii) there exists the greatest lower bound a ∧ b of a and b; (iii) there exists the least upper bound of a^{−1} and b^{−1} ; (iv) there exists the greatest lower bound of a^{−1} and b^{−1} .

In particular, if there exists a ∨ b, then for any c, d ∈ G we have

ca ∨ cb = c(a ∨ b), ad ∨ bd = (a ∨ b)d
a ∧ b = (a^{−1} ∨ b^{−1} )^{−1} , a ∧ b = a(a ∨ b)^{−1} b.

Thus, (G, ∨, ∧, ·) is a lattice ordered group.

Suppose that (S, ∨, ∧, ·) is a lattice ordered Clifford semigroup under the partial order ⩽. If ⩽ is amenable then (S, ∨, ∧, ·) is called an amenably lattice ordered Clifford semigroup. Thus we have

Theorem 3.2. Suppose that (S, ∨, ·) a ∨-semilatticed Clifford semigroup under the amenable partial order ⩽. Then (S, ∨, ∧, ·) is an amenably lattice ordered Clifford semigroup.

Proof. Suppose that a, b ∈ S. It is easy to see that (ab^{0}, ba^{0}) ∈ H. That is to say ab^{0}, ba^{0} ∈ H_{a^{0} b^{0}} . By Corollary 3.1, we have that H_{a^{0} b^{0}} is a ∨-semilatticed group, it follows from Lemma 3.2 that H_{a^{0} b^{0}} is lattice ordered group. For any x, y ∈ H_{a^{0} b^{0}} we denote by x ∧ y the great lower bound of x and y. Thus, there exists an element c ∈ H_{a^{0} b^{0}} is the great lower bound of ab^{0} and ba^{0}, i.e., c = b^{0}∧ba^{0}. Hence we have from ⩽ is amenable that c ⩽ ab^{0} ⩽ a and c ⩽ b.

Suppose that d ∈ S such that d ⩽ a, b. Then d^{0} ⪯ a^{0}, d^{0} ⪯ b^{0}. Thus we have that d = db^{0} ⩽ ab^{0}, d = da^{0} ⩽ ba^{0} and so d ⩽ ab^{0} ∧ ba^{0}, that is to say d ⩽ c. This shows that ab^{0} ∧ ba^{0} is the greatest lower bound of a and b. We denote by a ∧ b the greatest lower bound of a and b. Thus we have that a∧b = ab^{0} ∧ba^{0}. Furthermore, we have that

a ∧ b = ab^{0} ∧ ba^{0} = ((ab^{0})^{−1} ∨ (ba^{0})^{−1} )^{−1} = (a^{−1} ∨ b^{−1} )^{−1} a^{0}b^{0}.

In the following we will prove that (S, ∨, ∧, ·) is a lattice ordered Clifford semigroup. For any c ∈ S we have that

ac ∧ bc = [(ac)^{−1} ∨ (bc)^{−1} ]^{−1} (ac)^{0}(bc)^{0} = [c^{−1} (a^{−1} ∨ b^{−1} )]^{−1} a^{0}b^{0}c^{0}
= (a^{−1} ∨ b^{−1} )^{−1} ca^{0}b^{0}c^{0} = (a^{−1} ∨ b^{−1} )^{−1} a^{0}b^{0}c = (a ∧ b)c,

Dually, we have that

ca ∧ cb = [(ca)^{−1} ∨ (cb)^{−1} ]^{−1} (ca)^{0}(cb)^{0} = [(a^{−1} ∨ b^{−1} )c^{−1} ]^{−1} a^{0}b^{0}c^{0}
= c(a^{−1} ∨ b^{−1} )^{−1} a^{0}b^{0}c^{0} = cc^{0}(a^{−1} ∨ b^{−1} )^{−1} a^{0}b^{0} = c(a ∧ b).

This shows that (S, ∧, ·) is a ∧-semiltticed semigroup.

Since (S, ∨, ·) a ∨-semilatticed Clifford semigroup, it follows that (S, ∨, ∧, ·) is a lattice ordered Clifford semigroup. □

Suppose that (S, ∨, ∧, ·) is an amenably lattice ordered Clifford semigroup. It follows from Theorem 3.2 that (a ∧ b)^{0} = a^{0}b^{0} = a^{0} ∧ b^{0} since a ∧ b = ab^{0} ∧ ba^{0} and H_{a^{0} b^{0}} is a lattice ordered group. Assume that a, b ∈ S such that (a, b) ∈ H. For any c ∈ S, we have that (a ∨ c)^{0} = a^{0} ∨ c^{0} = b^{0} ∨ c^{0} = (b ∨ c)^{0}. This implies that (a ∨ c, b ∨ c) ∈ H and so H is a congruence on (S, ∨). Similarly, we can obtain that H is a congruence on (S, ∧). This shows that H is a congruence on S. It follows that S/H is a distributive lattice. Also, we can define a binary relation σ on S as follows:

(∀a, b ∈ S) (a, b) ∈ σ ⇔ (∃e ∈ E(S)) ae = be.

It follows from Proposition 5.3.2 in [8] that σ is the least group congruence on the multiplicative reduct of S. Assume that a, b ∈ S and that (a, b) ∈ σ. Then there exists e ∈ E(S) such that ae = be. For any c ∈ S, we have that ae ∨ ce = be ∨ ce. That is to say, (a ∨ c)e = (b ∨ c)e. This implies that (a ∨ c, b ∨ c) ∈ σ. Since (S, ∨) is a semilattice, we also have that (c ∨ a, c ∨ b) ∈ σ. This shows that σ is a congruence on (S, ∨). Similarly, we can obtain that σ is a congruence on (S, ∧). This shows that σ is a congruence on S. Thus we have that σ is a congruence on S. It follows that S/σ is a lattice ordered group.

Suppose that (S, ∨, ∧, ·) is an amenably lattice ordered Clifford semigroup. If the multiplicative reduct of S is E-unitary, then it follows from Corollary 4.3.6 in [15] that H ∩ σ = Δ. By Theorem 2.2, Corollary 4.3.6 in [15] and Theorem 3.5 in [21] we have

Theorem 3.3. Let (S, ∨, ∧, ·) be an amenably lattice ordered Clifford semigroup. Then the following statements are equivalent:

(i) S is a sturdy distributive lattice of lattice ordered groups; (ii) S is a subdirect product of a distributive lattice and a lattice ordered group; (iii) the multiplicative reduct of S is E-unitary; (iv) the multiplicative reduct of S is a sturdy semilattice of groups;

(v) the multiplicative reduct of S is a subdirect product of a semilattice and a group.

Remark 3.1. It is clear that both a lattice ordered Clifford semigroup and a lattice ordered group are algebras of type (2, 2, 2). Also, a distributive lattice can be considered as an algebra of type (2, 2, 2) whose multiplication and meet coincide. Thus Theorem 3.3 characterizes the amenably lattice ordered Clifford semigroup which can be expressed as a sturdy distributive lattice (as an algebra of type (2, 2, 2)) of lattice ordered groups.

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School of Mathematics (Received 31 12 2013) Northwest University Xian, P.R. China

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