Errata on “On the Variety of Heyting Algebras with Successor Generated by all Finite Chains

Castiglioni, José L.

San Martín, Hernán J.

Portal Czasopism Naukowych (E-Journals)

REPORTS ON MATHEMATICAL LOGIC
48 (2013), 117–120
DOI: 10.4467/20842589RM.13.006.1258
Jose´ L. CASTIGLIONI and Herna´n J. SAN MARTI´N
ERRATA ON “ON THE VARIETY OF
HEYTING ALGEBRAS WITH SUCCESSOR
GENERATED BY ALL FINITE CHAINS”
In [3] we have claimed that ﬁnite Heyting algebras with successor only
generate a proper subvariety of that of all Heyting algebras with successor,
and in particular all ﬁnite chains generate a proper subvariety of the latter.
As Xavier Caicedo made us notice, this claim is not true. He proved, using
techniques of Kripke models, that the intuitionistic calculus with S has
ﬁnite model property and from this result he concluded that the variety of
Heyting algebras with successor is generated by its ﬁnite members [2].
This fact particularly aﬀects Section 3.2 of our article. Concretely, in
Remark 3.3, our claim “Let K be a class of S-Heyting algebras of height
less or equal to a ﬁxed ordinal ξ. Using the categorical duality between S-
Heyting algebras and S-Heyting spaces, it can be shown that the elements
of classes H(K), S(K) and P(K) have also height less or equal to ξ. Here
H, S and P are the class operators of universal algebra. Hence for each
ordinal ξ, the class of S-Heyting algebras of height less or equal to ξ is a
variety“ is not true as stated. It remains valid only if ξ is a ﬁnite ordinal.
Received 13 September 2011
118 JOSE´ L. CASTIGLIONI AND HERNA´N J. SAN MARTI´N
In particular, the class of S-Heyting algebras of height ω is not a variety
and the variety generated by all ﬁnite chains is exactly the variety of linear
S-Heyting algebras.
In what follows, instead of using the proof given in [2], which is not
published, we shall give a simple algebraic proof that the variety of linear
Heyting algebras with S is generated by the ﬁnite chains.
Let T be the type of Heyting algebras with successor built in the usual
way from the operation symbols ∧, ∨, →, and S corresponding to meet,
join, implication and successor, respectively. Write T (X) for the term
algebra of type T with variables in the set X. It is well known that any
function v : X → H, with H a S-Heyting algebra, may be extend to a
unique homomorphism v : T (X) → H.
Write SLH for the variety of linear S-Hetying algebras. Recall that
SLH is said to have the ﬁnite model property (FMP) if for every ϕ ∈ T (X)
there is a linear S-Heyting algebra H and a homomorphism v : T (X) → H
such that if v(ϕ) 
= 1 then there is a ﬁnite linear S-Heyting algebra L and
a homomorphism w : T (X) → L such that w(ϕ) 
= 1. Let us prove that
SLH has the FMP. In so doing we shall use the two following well known
Lemmata.
Lemma 1. (Lemma 1.1 of [4]) If P is a prime ﬁlter in a linear algebra
H, then H/P is a chain.
Lemma 2. Let C be a S-Heyting algebra which is a chain and L a
bounded sublattice of C, endowed with its implication →L and successor
SL, as ﬁnite lattice. Then, we have that,
1. If x, y ∈ L then x → y = x →L y.
2. If x, S(x) ∈ L then SL(x) = S(x).
Take α and β in T (X). Note that an equation α ≈ β holds in a S-
Heyting algebra H if and only if α → β ≈ 1 holds in H; and the latter is
equivalent to ask that for any homomorphism v : T (X) → H, v(α → β) =
1.
We are now ready to prove the main result.
Proposition 3. The variety SLH has the FMP.
ERRATA 119
Proof. Let ψ ∈ T (X), H be a linear S-Heyting algebra and v : T (X) →
H be a homomorphism such that v(ψ) 
= 1. Let→ and S be the implication
and the successor of H respectively. We will ﬁnd a ﬁnite chain L and a
homomorphism t : T (X) → L such that t(ψ) 
= 1.
By the Prime Filter Theorem there is a prime ﬁlter P of H such that
v(ψ) /∈ P , so v(ψ)/P 
= 1. Write C in place of H/P . Hence, by Lemma
1, we have that C is a chain. On the other hand, using that the successor
operator is compatible [1] we have that the quotient function ρ : H → C is
a homomorphism. Hence w = ρv : T (X) → C is a homomorphism. Note
that w(ψ) = v(ψ)/P 
= 1.
Let Subψ = {ψ1, ..., ψn} be the set of subformulas of ψ and L the subset
of C given by {0, 1} ∪ {w(α) : α ∈ Subψ}. If V is the set of propositional
variables which appear in ψ, we can deﬁne the function t : X → L in the
following way:
t(xi) =
{
w(xi) if xi ∈ V
0 if xi /∈ V .
We know that t may be uniquely extended to a homomorphism t :
T (X) → L. Using Lemma 2, we can prove, by an easy induction on
sentences, that : t(ψi) = w(ψi) for i = 1, ..., n. Therefore we have that
t(ψ) = w(ψ) 
= 1. 
In particular, we have that the following corollary holds.
Corollary 4. The variety SLH is generated by all ﬁnite chains.
Finally, we want to call the attention to a typos in Proposition 5.6. We
wrote SHS in place of SLH.
.References
[1] X. Caicedo, R. Cignoli, An algebraic approach to intuitionistic connectives. Journal
of Symbolic Logic 66:4 (2001), pp. 1620-1636.
[2] X. Caicedo, Kripke semantics for Kuznetsov connective, Personal comunication
(2008).
[3] J.L. Castiglioni, H.J. San Mart´ın, On the variety of Heyting algebras with successor
generated by all ﬁnite chains. Reports on Mathematical Logic 45 (2010), pp. 225-248.
120 JOSE´ L. CASTIGLIONI AND HERNA´N J. SAN MARTI´N
[4] A. Horn, Logic with truth values in a linearly ordered Heyting algebra, J. Symbolic
Logic 34 (1969), pp. 395-408.
Departamento de Matematica,
Facultad de Ciencias Exactas, UNLP.
Casilla de correos 172,
La Plata (1900)
Argentina
jlc@mate.unlp.edu.ar
hsanmartin@mate.unlp.edu.ar


Errata on “On the Variety of Heyting Algebras with Successor Generated by all Finite Chains”

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Errata on “On the Variety of Heyting Algebras with Successor Generated by all Finite Chains”

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