A note on chain‐based semi‐Heyting algebras. Issue 4 (16th January 2021) - Cornejo, Juan Manuel
Abstract: 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 m th 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 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.