A note on regular De Morgan semi-Heyting algebras

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains RDMSH1 . Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation.


Introduction
Semi-Heyting algebras were introduced by us in [12] as an abstraction of Heyting algebras. They share several important properties with Heyting algebras, such as distributivity, pseudocomplementedness, and so on. On the other hand, interestingly, there are also semi-Heyting algebras, which, in some sense, are "quite opposite" to Heyting algebras. For example, the identity 0 → 1 ≈ 0, as well as the commutative law x → y ≈ y → x, hold in some semi-Heyting algebras. The subvariety of commutative semi-Heyting algebras was defined in [12] and is further investigated in [13].
Key words and phrases. regular De Morgan semi-Heyting algebra of level 1, lattice of subvarieties, amalgamation property, discriminator variety, simple, directly indecomposable, subdirectly irreducible, equational base.
Heyting algebras (see [8]) so that we could settle an old conjecture of ours.
The concept of regularity has played an important role in the theory of pseudocomplemented De Morgan algebras (see [9]). Recently, in [15] and [16], we inroduced and examined the concept of regularity in the context of DQDSH and gave an explicit description of (twenty five) simple algebras in the (sub)variety DQDStSH 1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. The work in [15] and [16] led us to conjecture that the variety RDMSH 1 of regular De Morgan algebras satisfies Stone identity.
The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH 1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity, thus settlieng the above mentioned conjecture affirmatively. As applications of this result and the main theorem of [15], we present (equational) axiomatizations for several subvarieties of RDMSH 1 . Secondly, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH 1 of regular dually quasi-De Morgan Stone semi-Heyting algebras, of which RDMSH 1 is a subvariety. Furthermore, we prove that every subvariety of RDQDStSH 1 , and hence of RDMSH 1 , has Amalgamation Property. The note concludes with some open problems for further investigation.

Dually Quasi-De Morgan Semi-Heyting Algebras
The following definition is taken from [12]. An algebra L = L, ∨, ∧, →, 0, 1 is a semi-Heyting algebra if L, ∨, ∧, 0, 1 is a bounded lattice and L satisfies: Let L be a semi-Heyting algebra and, for x ∈ L, let x * := x → 0. L is a Heyting algebra if L satisfies: L is a commutative semi-Heyting algebra if L satisfies: (Co) x → y ≈ y → x. L is a Boolean semi-Heyting algebra if L satisfies: (Bo) x ∨ x * ≈ 1. L is a Stone semi-Heyting algebra if L satisfies: Semi-Heyting algebras are distributive and pseudocomplemented, with a * as the pseudocomplement of an element a. We will use these and other properties (see [12]) of semi-Heyting algebras, frequently without explicit mention, throughout this paper.
The following definition is taken from [14].
If the underlying semi-Heyting algebra of a DQDSH-algebra is a Heyting algebra we denote the algebra by DQDH-algebra, and the corresponding variety is denoted by DQDH.
In the sequel, a ′ * ′ will be denoted by a + , for a ∈ L ∈ DQDSH. The following lemma will often be used without explicit reference to it. Most of the items in this lemma were proved in [14], and the others are left to the reader.
Next, we describe some examples of DQDSH-algebras by expanding the semi-Heyting algebras given in Figure 1. These will play a crucial role in the rest of the note.
Let 2 e and2 e be the expansions of the semi-Heyting algebras 2 and 2 (shown in Figure 1) by adding the unary operation ′ such that 0 ′ = 1, Let L dp i , i = 1, . . . , 10, denote the expansion of the semi-Heyting algebra L i (shown in Figure 1) by adding the unary operation ′ such that 0 ′ = 1, 1 ′ = 0, and a ′ = 1.
We Let C dp 10 := {L dp i : i = 1, . . . , 10} and C dm 10 := {L dm i : i = 1, . . . , 10}. We also let C 20 := C dm 10 ∪ C dp 10 . Each of the three 4-element algebras D 1 , D 2 and D 3 has its lattice reduct as the Boolean lattice with the universe {0, a, b, 1}, b being the complement of a, has the operation → as defined in Figure 1, and has the unary operation ′ defined as follows: The following is a special case of Definition 5.5 in [14]. Let x ′ * ′ * := x 2(′ * ) . Note that x 2(′ * ) ≤ x in a DMSH-algebra. DEFINITION 2.3. The subvariety DMSH 1 of level 1 of DMSH is defined by the identity: x ∧ x ′ * ∧ x 2(′ * ) ≈ x ∧ x ′ * , or equivalently, by the identity: It follows from [14] that the variety DMSH 1 , is a discriminator variety. We note here that the algebras described above in Figure 1 are actually in DMSH 1 .
Then L is regular if L satisfies the following identity: (R) x ∧ x + ≤ y ∨ y * . The variety of regular DMSH 1 -algebras will be denoted by RDMSH 1 .
Proof. Let a ∈ L. If a * = 0, Then the theorem is trivially true. So, we can assume that a * = 0. Then a ∨ a * = 1, in view of the preceding lemma. The conclusion is now immediate.

Applications
Let V(K) denote the variety generated by the class K of algebras. The following corollary is immediate from Theorem 3.7 and Corollary 3.4(a) of [16], and hence is an improvement on Corollary 3.4(a) of [16].
, L dm 10 , D 1 ). Proof. It suffices, in view of (a) of the preceding corollary, to verify that L dm 9 , L dm 10 , and D 1 satisfy the pseudocommutative law, while the rest of the simples in RDMSH 1 do not.
The proofs of the following corollaries are similar.
COROLLARY 4.18. Each of the following identities is an equational base for V(D 2 ), mod RDMH 1 : V(D 1 ) was axiomatized in [14]. Here are more bases for it.
COROLLARY 4.19. Each of the following identities is an equational base for V(D 1 ), mod RDMcmSH 1 :

Lattice of subvarieties of RDQDStSH 1
We now turn to describe the lattice of subvarieties of RDQDStSH 1 which contains RDMSH 1 in view of Theorem 3.7. For this purpose we need the following theorem which is proved in [15]. (1) L is simple Let L denote the lattice of subvarieties of RDQDStSH 1 . T denotes the trivial variety, and, for n a positive integer, B n denotes the natom Boolean lattice. We also let 1 + B denote the lattice obtained by adding a new least element 0 to the Boolean lattice B. : i = 5, 6, 7, 8}∪{L dp i : i = 5, 6, 7, 8}∪{D 3 }. Observe that each of the simples in S 1 contains 2 e . Let us first look at the interval [V(2 e ), V(S 1 )]. Since each algebra in S 1 is an atom in this interval, we can conclude that the interval is a 9-atom Boolean lattice; thus the interval [T, V(S 1 )] is isomorhic to 1 + B 9 . Similarly, since each of the simples in S 2 contains2 e , it is clear that the interval [T, V(S 2 )] is isomorphic to 1 + B 5 . Likewise, since each of the simples in S 3 has only one subalgebra, namely the trivial algebra, the interval [T, S 3 ] is isomprphic to B 9 . Observe that the the intersection of the subvarieties V(S 1 ), V(S 2 ) and V(S 3 ) is T and their join is RDQDSH 1 in L. It, therefore, follows that L is isomorphic to (1 + B 9 )×(1 + B 5 )×B 9 .  Similar formulas can be obtained for other subvarieties of RDQDSH 1 .

Amalgamation
We now examine the Amalgamation Property for subvarities of the variety RDQDStSH 1 . For this purpose we need the following theorem from [3]. THEOREM 6.1. Let K be an equational class of algebras satisfying the Congruence Extension Property, and let every subalgebra of each subdirectly irreducible algebra in K be subdirectly irreducible. Then K satisfies the Amalgamation Property if and only if whenever A, B, C are subdirectly irreducible algebras in K with A a common subalgebra of B and C, the amalgam (A; B, C) can be amalgamated in K. THEOREM 6.2. Every subvariety of RDQDStSH 1 has the Amalgamation Property.
Proof. It follows from [14] that RDQDStSH 1 has CEP. Also, it follows from Theorem 5.1 that every subalgebra of each subdirectly irreducible (= simple) algebra in RDQDStSH 1 is subdirectly irreducible. Therefore, in each subvariety V of RDQDStSH 1 , we need only consider an amalgam (A : B, C), where A, B, C are simple in RDQDStSH 1 and A a subalgebra of B and C. Then it is not hard to see, in view of the description of simples in RDQDStSH 1 given in Theorem 5.1, that (A : B, C) can be amalgamated in V.