Central Elements in Varieties with the Fraser–Horn Property

✍ Scribed by Diego Vaggione

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 149 KB
• Volume: 148
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1855

✦ Synopsis

Combining [14, Lemma 3] with [8] it can be proved that the following conditions for a variety V are equivalent:

(1) No nontrivial algebra of V has a trivial subalgebra.

(2) There exist unary terms 0 1 (w), ..., 0 N (w), 1 1 (w), ..., 1 N (w) such that V< 09 (w)=19 (w) Ä x= y, where 09 =(0 1 , ..., 0 N ) and 19 =(1 1 , ..., 1 N ). 2

A variety satisfying (2) will be called a variety with 09 and 19 . If * # A # V, then we say that eÁ # A N is a *-central element of A if there exists an isomorphism A Ä A 1 _A 2 such that * Ä (* 1 , * 2 ) eÁ Ä ((0 1 (* 1 ), 1 1 (* 1 )), ..., (0 N (* 1 ), 1 N (* 2 ))).