ACTA issues

Homomorphisms and principal congruences of bounded lattices I. Isotone maps of principal congruences

G. Gr├Ątzer

Acta Sci. Math. (Szeged) 82:3-4(2016), 353-360

Abstract. Two years ago, I characterized the order $\Princl L$ of principal congruences of a bounded lattice $L$ as a bounded order. If $K$ and $L$ are bounded lattices and $\gf $ is a \zo homomorphism of $K$ into $L$, then there is a natural isotone \zo map $\Princl\gf $ from $\Princl K$ into $\Princl L$. We prove the converse: For bounded orders $P$ and $Q$ and an isotone \zo map $\gy $ of $P$ into $Q$, we represent $P$ and $Q$ as $\Princl K$ and $\Princl L$ for bounded lattices $K$ and $L$ with a \zo homomorphism $\gf $ of $K$ into $L$, so that $\gy $ is represented as $\Princl\gf $.

DOI: 10.14232/actasm-015-056-y

AMS Subject Classification (1991): 06B10

Keyword(s): bounded lattice, congruence, principal, order

Received July 20, 2015, and in revised form September 18, 2015. (Registered under 56/2015.)