The variety generated by semi-Heyting chains

2011

Stud Logica

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [48] and [50] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH. We also use it to prove that the variety DQSSH of dually quasi-Stone semiHeyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH. Finally, we apply these results to give bases for "small" subvarieties of BDQDSH, DQSSH, and DblSH.

“…Hence (1) follows from Theorem 8.1. From (1) and Theorem 9.4 of [19] we conclude (2). For (3), we have from Theorem 5.4 that DQSSH is arithmetical.…”

Section: Semi-heyting Algebras With a Dual Quasi-stone Operationmentioning

confidence: 76%

“…In fact, it can be easily seen that if f (n) = the number of semi-Heyting chains, then (f (n)) 2 is the number of double semi-Heyting chains. See [2] for a formula to calculate f (n).…”

Section: Double Semi-heyting Algebrasmentioning

confidence: 99%

Expansions of Semi-Heyting Algebras I: Discriminator Varieties

2011

Stud Logica

“…Moreover, there is a rich supply of algebras in SH. It is known that there are in SH, up to isomorphism, two 2-element algebras, ten 3-element algebras, only one of which, of course, is a Heyting algebra and 160 algebras on a 4-element chain (see [38] and [4]).…”

Section: Conjecturementioning

confidence: 99%

“…There exists already some literature related to this problem. The papers that deal with this problem algebraically include [38], [2], [3], [4], [5], [15] and [17]. The paper [4] investigates the properties of semi-Heyting chains and the structure of the variety CSH generated by all semi-Heyting chains.…”

Section: Conjecturementioning

confidence: 99%

“…The papers that deal with this problem algebraically include [38], [2], [3], [4], [5], [15] and [17]. The paper [4] investigates the properties of semi-Heyting chains and the structure of the variety CSH generated by all semi-Heyting chains. In [2], it is proved, among other things, that the variety of Boolean semi-Heyting algebras (algebras with an underlying structure of Boolean lattice) constitutes a reflective subcategory of SH, extending the corresponding result for Heyting algebras (see [6,Corollary IX.5.4], and that the free algebras in a subvariety V of SH are directly indecomposable if and only if V satisfies the Stone identity, extending a known result for Heyting algebras.…”

Section: Conjecturementioning

confidence: 99%

See 1 more Smart Citation

Semi-Heyting Algebras and Identities of Associative Type

Cornejo

2019

B Sect Log

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. 𝒮ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras. They share several important properties with Heyting algebras. An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of 𝒮ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety 𝒮ℋ of asociative type of length 3. Our main result shows that there are 3 such subvarities of 𝒮ℋ.

A note on chain‐based semi‐Heyting algebras

Cornejo

Monteiro

Mathematical Logic Qtrly

et al. 2020

We determine the number of non-isomorphic semi-Heyting algebras on an n-element chain, where n is a positive integer, using a recursive method. We then prove that the numbers obtained agree with those determined in [1]. We apply the formula to calculate the number of non-isomorphic semi-Heyting chains of a given size in some important subvarieties of the variety of semi-Heyting algebras that were introduced in [5]. We further exploit this recursive method to calculate the numbers A(n, m) of non-isomorphic semi-Heyting chains with n elements such that removing the mth element (1 < m < n) we are left with a subalgebra. We also solve a related problem posed in [1] of determining the number of ways a semi-Heyting chain with n − 1 elements can be extended to a n element semi-Heyting chain by adding a new element in the mth place. Finally we combine these results by finding a second way to calculate the numbers A(n, m) that provides some extra information.