A note on chain-based semi-Heyting algebras

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

MATHEMATICAL LOGIC QUARTERLY

WILEY-V C H VERLAG GMBH

Lugar: Weinheim; Año: 2020 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 [ACDV10]. 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 [San08]. 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 m-th element (1 < m < n) we are left with a subalgebra. We also solve a related problem posed in [ACDV10] 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 m-th place. Finally we combine these results by finding a second way to calculate the numbers A(n, m) that provides some extra information.