A note on chain-based semi-Heyting algebras

CORNEJO, JUAN MANUEL; MONTEIRO, LUIZ F.; SANKAPPANAVAR, HANAMANTAGOUDA P.; VIGLIZZO, IGNACIO D.

MATHEMATICAL LOGIC QUARTERLY

WILEY-V C H VERLAG GMBH

Año: 2021 vol. 66 p. 409 - 417

0942-5616

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 (Formula presented.) of non-isomorphic semi-Heyting chains with n elements such that removing the mth element ((Formula presented.)) 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 (Formula presented.) 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 (Formula presented.) that provides some extra information.