Antiatomic retract varieties of monounary algebras
Danica Jakubı́ková-Studenovská · Czechoslovak Mathematical Journal · 1998
Retracts of monounary algebras were investigated in the papers [2]-[4]. The notion of the retract variety of monounary algebras was introduced in [5] by applying an analogy with the notion of the order variety of partially ordered sets studied in [1]. The collection R of all retract varieties of monounary algebras was investigated in [5]. This collection is considered to be partially ordered by the class-theoretical inclusion. A retract variety y is called atomic if ^ 0 and, whenever V is a retract variety with 0 ^ V C V, then y = V. It was proved that there are exactly 2° atomic retract varieties in JH. A retract variety Y of 91 is said to be antiatomic if I/ ^ 0 and there is no atomic variety Vi of £K with YI C V. In view of the relation C for pairs of retract varieties V\, % of 91, we apply also the symbols inf {Vi,%} and sup {/l,^} in the usual way. Namely, if %, %, % belong to 5H and (i) ?i C y3, Y2 C ys, (ii) if V belongs to JR, ?i C y, % C Y, then y3 C y, then we write /s =sup{'^i,^2}. The notion inf {^i//^} is defined dually. The description of all antiatomic retract varieties of 9t is given in Theorem 3.2. Further we investigate the collection Ant of all antiatomic retract varieties of 5H; the following results will be proved: (a) Ant is closed with respect to the operations of inf and sup. (b) There is a proper class ff of ordinals with the following properties: (bl) For each a € ff there exists Wa € Ant such that, whenever a,0 £ ff, a^/3,thenWa <£%.