Monatsh Math (2011) 164:23–37 DOI 10.1007/s00605-010-0225-9 Remarks on homomorphism-homogeneous lattices and semilattices Igor Dolinka · Dragan Mašulovi´ c Received: 16 March 2010 / Accepted: 15 April 2010 / Published online: 5 May 2010 © Springer-Verlag 2010 Abstract We consider lattices and semilattices enjoying the homomorphism-homo- geneity property introduced recently by P. J. Cameron and J. Nešetˇ ril. First we com- pletely characterize all homomorphism-homogeneous lattices. Also, as a consequence of some general results, we exhibit transparent examples of semilattices both with and without this property. Finally, we show that the endomorphism monoid of , the (unique) countable universal homogeneous semilattice, embeds every finite semi- group. Keywords Homomorphism-homogeneity · Lattice · Semilattice Mathematics Subject Classification (2000) 06A12 · 06B05 · 03C07 1 Introduction The motivation of this note traces back to the seminal paper of Cameron and Nešetˇ ril [9], which initiates the study of homomorphism-homogeneous first-order structures. This notion is closely related to and inspired by—although not a generalization of— the classical notion of homogeneity (or ultrahomogeneity) from model theory [15]. Supported by the Ministry of Science and Technological Development of the Republic of Serbia, Grant Nos. 144011 and 144017, respectively. I. Dolinka (B ) · D. Mašulovi´ c Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovi´ ca 4, 21000 Novi Sad, Serbia e-mail: dockie@dmi.uns.ac.rs D. Mašulovi´ c e-mail: masul@dmi.uns.ac.rs 123