DEM0NSTRAT10 MATHEMATICA VU. XVI No 4 IN) Ewi Graczyúska PROOFS OF UNARY REGULAR IDENTITIES The notion of regular identity was introduced by J.Pionka in [ej. Varieties defined by regular identities were consider- ed in [2j, [3], [5] and [l"0 . Given a variety K, H(K) denotes the set of all regular identities satisfied in K. In [3J we considered the problem of bases for varieties defined by R(k) if K is strongly non- regular variety (see C2J). in this note we deal with ana- logous problems for unary algebras. Notation Our terminology is basically that of [4J, DÜ* consi- der unary varieties i.e. we deal with algebras Ot= (A,(f t :teT)) of type r: T —- {l}  For a given variety V of type r, ¿(V) denotes the lattice of all subvarieties of V. By p(x), <7 (x) etc. etc., we denote unary polynomial symbols (on a variable x). R(r) denotes the.,set of all regular identities of type r, i.e. identities of the form p(x) = q(x) (see [V]). Por a given set E of identities, denotes the variety of type r de- fined by . E(K) denotes the set of all identities satisfied in K and R(K) = B(K)D R(r). Let e be an identity of type r . Following £7], [ll] , if we assume that x = x e 22 and 22 is closed under forming sub- stitutions, then by a "derivation" we mean a sequence (e 1 ,...,e n ) such that for i = 1,...,n-1 there exists an iden- tity ( ^ = yS^e 22 such that «j^ (or ft^) i s a subterm! of e.^ - 925 - Unauthenticated Download Date | 3/4/20 5:36 PM