Chvalina, Jan

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On connected unars with regular endomorphism monoids

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJan ChvalinaOn connected unars with regular endomorphism monoidsArchivum Mathematicum, Vol. 16 (1980), No. 1, 7--13Persistent URL: http://dml.cz/dmlcz/107051Terms of use:© Masaryk University, 1980Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCH. MATH. 1, SCRIPTA FAC SCI. NAT. UJEP BRUNENSIS XVI: 7—14, 1980 ON CONNECTED UNARS WITH REGULAR ENDOMORPHISM MONOIDS JAN CHVALINA, Brno (Received March 3, 1979) A monounary algebra, i.e. a pair (A9f)9 where A is a non-void set andfa self-map of the set A9 is briefly called a unar. This paper aims to give some conditions of the topological and algebraic character equivalent to the regularity of the endomorphism monoid of a connected unar. There are used the descriptions of unars with regular and inverse endomorphism monoids obtained by L. A. Skornjakov in [12] and results of papers [4], [6]. In the below stated characterizations we consider mostly endomorphism monoids which are not groups. For the characterization of unars whose endomorphism monoids are automorphism groups see [12] Theorem 3. Fundamental used notions concerning monounary algebras can be found e.g. in papers [5], [8], [11], [12]. Let (A9f) be a connected unar. The set of all cyclic elements of (A9f) (i.e. such elements as A that fH(a) «• a for some integer n *z 1) will be denoted in regard with [8] by A*2 and further A001 = {x e A \ Ami: there is a sequence {xt}i€m such that x0 » x and f(xt+t) = xt for each ieco}9 A0 ** =* {x e A:f~l(x) = 0}. A unar is called a cycle if A « Ami. The upper cone of an element a, i.e. the set {/"(a): n = 0,1,2,...} will be denoted by [a)/, the lower cone {xeA: fn(x) = a for some neco} by (a]f. We agree on denoting the cardinality of a set A by | A |. A connected unar (A9 f) with | A | = K0 andf - a permutation of A is called a line. A connected unar (A9f) is said to be a cycle with short tails or a line with short tails if it contains a cycle or a line C such thatf(*) e C for every x e A. If | Bmi | £ 1 for each component (B9 fB) of a unar (A9f) we put a jg ft for a9 b e A if there exists n e a> withf*(a) == b and a < fb if a jg fb9 a # b. Further, we denote by (A, f) the factor-unar (i.e. the factor-algebra of a monounary algebra (A9f)) corresponding to the congruence mf on (A9f) defined by a zsfb if a » b or a.beA*32. The monoid of all endomorphisms of (A9f) is denoted by E(A9f). For the definition of a regular and inverse semigroup see [3] § 1.9. A certain strengthening of the notion of a regular semigroup is the notion of an anti-regular semigroup (cf. [10]) called in [1] an anti-inverse semipoup* Let us recall the necessary definitions (see [1] and [10]): A semigroup S is said to be anti-inverse if for each element aeS there is an element be S such that aba = b and bob = a. The elements a and b are then called anti-inverses. A saturated topological space called also quasi-discrete ([2] 26A) is a topological space (A> t) with the completely additive topological closure operation % i.e. each point of this space possesses the minimum neighbourhood (cf. [9]). A discrete space of Alexandrov is a saturated T0-space. Compactness is meant in the sense of [2] 41A, i.e. quasi-compactness considered in [9]. A continuous closed self-map of a topological space (A, T) will be called as usual a closed deformation of (A91) and the monoid of all closed deformations of this space will be denoted by S(A91) We say that a topological space (A9x) has the fixed set property or briefly the FS-property (the fixed point property, briefly the FP-property) with respect to closed deformations if there exists a non-void proper subset J c i (a point xe A) with f(X) = X(f(x) = x) for each/e S(A9 T). In what follows g means the usual set inclusion and A c B means A g B A* B. Theorem 1. Let (A9 f) be a connected unar whose endomorphism monoid is not a group. Then E(A9f) is regular if and only if there exists a discrete topology of Alexandrov % on the set A such that E(A9f) = S(A91) and the space (A91) has the FS-property with respect to closed deformations. Proof. Let (A9f) be a connected unar satisfying the assumption of the theorem. Since (A, f) contains at most one cyclic element, by Theorem 3.3 [4] there exists a discrete topology of Alexandrov r with E(A9 f) = S(A, t) if and only if the unar (A, f) has one of the following forms: (0 f2 = f, (ii) A -« Ami u A°9 where either A0 = 0 or (Am\ S/) is a chain of the type co* © co and A0 = 0 (i.e. (A9f) is a line with short tails), (iii) A = A0 u Ax, where (Al9 £f) is a chain of the type co with the first element c and f(a) « c for each a e A0. Suppose A = Aa0i and simultaneously (At001,/) is not a line. Admit there exists a non-void set B a A with g(B) = B for each g e E(A9 f). Since/* € E(A9f) for every k e co9 the ordered set (B9 S/) does not contain any minimal and maximal element and [A)/ g B for each b e B. There exists a pair of elements a9 b e A such that aeA\B9 beB and fn(a) = f*(b) for some new. Since elements a,b form a pair of h-elements in the sense of [8] Definition 1.22 and xii [8] there exists g e E(A9f) such that g(b) = a. We get a contradiction, hence in the considered case for every non-void subset Be A there exists an endomorphismg of (A9f) withg(B) # B. Consequently (At001,/) is a line in the considered case. Since the existence of a non-void subset B g A with the property g(B) = B for each g e E(A, f) implies the inclusion B g g Am% u Am2 we have that the case (iii) is eliminated. On the other hand if (Aff) is a connected unar with/2 = /and \A\ ^ 2 or (A9f) is a line with short tails then A contains an lT(^,/)-invariant non-void proper subset. (A singleton formed by the cyclic element in the first case and the carrier of the line in the second one). Therefore 8 there exists a discrete topology of Alexandrov T oH A with S(A, f) « E(A9 f) and the space (A, T) has the FS-property with respect to closed deformations if and only if (A9f) is either a cycle with short tails or a line with short tails. Now, from Theorem 1 [12] there follows the assertion, q.e.d. In the following proposition LT(A) means the left zeros subsemigroup of the full transformation monoid T(A) on the set A. Recall that a unar is said to be nested if the system of all its subunars ordered by means of set inclusion forms a chain. Proposition 1. Let (A9f)bea connected unar. The following conditions are equivalent: 1 ° E(A9f) is regular and LT(A) n E(A9fk) # 0 for some kern. 2° There exists a compact saturated topology x on A with the property E(A91) = = 5(A, T). 3° There exists a saturated topology x on A with E(A9 f) = 5(A, T) and the space (A, T) has the FP-property with respect to closed deformations. Proof. l°-=>2°: Since for some positive integer kem there exists a constant self-map g of A with g e E(A9fk) we have by [12] Theorem 1 (A9f) is a cycle with short tails (or without tails). If we define a topology x on the set A by putting a T-C1O-sure of a subset X c A as xX = X u f(X)9 Condition 2° is satisfied. 2° => 3°: Let x be a compact saturated topology on the set A such that E(A91) « = S(A, T). By [4] Theorem 3.3 the unar (A, f) has one of the forms (i)-(iii) listed in the proof of Theorem 1. For each aeA there exists a nested subunar (B9fB) of (A9f), an element beB and a surjective homomorphism g: (A9f) -• (B9fB) such that g(a) = b and the equality fm(a) = fn(b) with integers m9 n minimal with respect to this property implies m = n. Since tn e S(A, T) for each n e co we have that for each aeA the closure x{a} is a right cofinal subset of [a)f and has the following property: If x9y9ze x{a}9 x <ty <tz then from fn(x) = y9 im(y) = z with minimal m, n it follows either n = m or z = t(y). Then the least T-neighbourhood of a (i.e. the closure of {a} in the saturated topology dual to T) is a left cofinal subset of (d}(. Since the space (A, T) is compact by [9] Proposition 1 the unar (A, f) contains a cyclic element, say e. Hence f2 = f and g(e) = e for each g e S(A9 x). 3°=> 1°: Since feS(A9 x) and the unar (A, f) is connected there exists exactly one element eeA with i(e) = e. By [4] Theorem 3.3 f2 = f. Condition 1° follows easy with respect to [12] Theorem 1, q.e.d. Corollary. Let (A9f) be a connected unar. The following conditions are equivalent: 1° Each element ofE(A9f) has a unique anti-inverse element in E(A9f). 2° The set A is either a singleton or (A9 x) is the Sierpinski-space for each topology x on A with the property S(A9 x) = E(A9f). 3° There exists a discrete topology of Alexandrov x on A such that (A9 x) is a tower space and E(A9f) = S(A9 x). 4° There exists a saturated tower topology x on A such that the space (A9 x) has the FP-property with respect to closed deformations and E(A9f) = S(A9 T). 9 Proof follows immediately from Theorem 1, Proposition 1 and [10] Theorem 2. The other characterization is expressed in terms of the groupoid theory. Similarly to [5] § 1 we associate a groupoid with every connected unar (A,f). For a, be A, denote by m9 n the smallest non-negative integers such that fn(a) e \b)f9 fm(b) e [a)f. We put S(a9 b) = m — n. Evidently S(a9 b) -F S(b9 a) = 0 for each pair a9beA and S(a9 b) = S(b9 a) = 0 for each pair a, be A™2. Further we put aefb = f(b) if S(a9 b) <£ 0 and aefb = f(a) if S(a9 b) < 0. It is to be noted that the groupoid (A9 ef) associated in this way with a unar (A9f) is neither associative nor commutative in general. In papers [5], [6] the binary operation ef is denoted by Vf. The following statement is contained in [5] Proposition 1.2. Proposition 2. Let (A9f) be a connected unar such that either A002 = 0 or f2 = / ThenE(A9f)~E(A9ef). Proposition 3. Let (A9f) be a connected unar with the regular endomorphism monoid E(A9f). Then there exists a commutative binary operation o on the set A such that E(A9f)~E(A9o). Proof. According to [12] Theorem 1 the unar (A, f) has one of the follow-ing forms: (1) it is trivial (i.e. | A | = 1), (2) f2 = f, (3) (A, f) is a line, (4) (A, f) is a line with short tails. If one of cases (3), (4) occurs, then A = A, f = / . Putting for each pair a,beA:aob = = aej-b, we get evidently a commutative groupoid (A, o) satisfying the condition £(A, f) = E(A, o) with respect to Proposition 2, q.e.d. By an ideal of a groupoid (A,.) we mean a both-side ideal, i.e. a non-empty subset / £ A such that a e / , be A implies a .be J and b .aeJ. The principal ideal generated by an element a is denoted by J(a). An ideal / is said to be trivial if | /1 = 1. If (A,.) is a groupoid and / an ideal of this groupoid then the corresponding Rees factor-groupoid is denoted by (A/J, .j); cf. [3] and [7]. A groupoid (A,.) is called distributive if for each triad a,b,ceA equalities a.(b.c) = (a.b). (a.c), (a.b). c = = (a.c). (b.c) hold and it is called a BD-groupoid (in accordance with [7]) if it satisfies one of the following equivalent conditions (see [7] Proposition 1.2): (i) (Af.) is distributive and the set of all its idempotents contains just one element, (ii) there is an element eeA with a.e = e = e.a (a zero of (A,.)) and a.(b.c) = = e = (a.b).c for all a,b,ceA. Theorem 2. Let (A,f) be a connected unar such that the endomorphism monoid E(A9f) is not a group. E(A,f) is regular if and only if (A, %) is a commutative groupoid containing the least proper ideal J with the following properties: 10 (i) The factor-groupoid (A / / , ej) is a BD-groupoid. (ii) If J is principal or contains an idempotent then it is trivial. Proof. Let (A9f) be a connected unar with the regular eftdomorphism monoid E(A9f) beifig not a group. With respect to [12] Theorem 1 and the definition of the binary operation ef we have that (A, %) is a commutative groupoid. If f2 -* f we put / = {e}, where e =* A002. If (A9f) is a line with short tails, i.e. A « -4001 u A0 with (Am%9f) a line-we put / « A*1. It can be easily shown (see the third part of the proof of Theorem 3.8 [6]) that in this case / i s the least proper ideal of the groupoid (A, ef). Since the Rees factor-groupoid (A/J, ef) of the groupoid (A, af) is associated with a connected idempotent unar (A//, F) (which is a factor unar of (A, f)) we have by [5] Lemma 1.3 that (A//, ej) is a BD-groupoid. The ideal / is principal if it contains an idempotent of (A, ef), i.e. if / =- {e}, where e is the only cyclic element of(A,f). Suppose 04,/) is a connected unar such that E(A9f) is not a group and such that (A, ef) is a commutative groupoid the least proper ideal / of which satisfies the above assumptions. From the commutativity of ef it follows that for each pair a9beA9 the equality 8(a9 b) =- 0 implies ((a) = f(b). Since E(A9f) is not a group, (A9f) is neither a cycle nor a line. If A*01 ^ 0 then it can be easily verified (in the same way as in the proof of Theorem 3.8 [6] p. 150) that the least ideal of (A, e§) coincides with the least subunar of (A, f) containing the set A°°'(-= Ami). This ideal is non-trivial hence (A9f) is a line with short tails. If A™% » 0 then there exists an element a e A with 5(a9 x) ^ 0 for each x e A. Then {fk(a): k = 1, 2,...} is the least ideal of (A, ef) and since it is principal we have f2 -= f. Hence (A9f) is a cycle with short tails. Apply-ing [12] Theorem 1 we get E(A9f) is regular, q.e.d. In [12] Theorem 2 there are given necessary and sufficient conditions under which the endomorphism monoid of a unar is an inverse semigroup. In fact these conditions strengthen those which are necessary and sufficient for the regularity of E(Aff). In the case of a connected unar E(A9f) is an inverse semigroup if and only if \f~~1(d) | S 2 for each a e A and (A9f) is either a cycle with short tails or a line with short tails (cf. [12] Theorem 2). From this result, using the binary operation ef9 we get the below stated characterization analogical to Theorem 3.9 [6]. For each element a of a groupoid (A9.) we put ^Ja -* {x e A : x.x » a}. Every element b€y/a is called a square root of the element a in the groupoid (A9.). If I sja | = 1 we say the element a possesses the unique square root in (A».). Especially, y/a ~fi(a) for each element a of the groupoid (A9 %) and thus evidently E(A9f) is a group (in the case of connected (A9f)) if and only if each element of (A9 ef) possesses the unique square root. Proposition 4. Let (A9f) be a connected unar. E(A9f) is an inverse semigroup if and only if either | sja | = 1 holds for each element a of (A9 ef) or \ y/a | <; 2 for 11 every ae(A, ef) and (A, sf) contains the least ideal J each element of which possesses the unique square root in (J, Bf). Proof. Let (A,f) be a connected unar. For E(A,f) being a group the assertion is evident. Thus we assume E(A,f) is not a group. If E(A,f) is an inverse semigroup then by [12] Theorem 2 for each a 6 At we have \f~l(a) \ = 2 and either A = A0 u u At001 (where (A^^f) is a line) or A = A0 u A™2. Putting J = A™' in the first case and / « At003 in the second one, we obtain the assertion with respect to A0 j&0. Assume the groupoid (A, ef) is satisfying conditions from the above proposition. Since each element of J possesses the unique square root in (/, ef) and yfa = f1(d) for each a e A, the subunar (J9fj) is either a cycle or a line. Since / is the least ideal of (A, sf) we have A \ J = A0. Thus (A,f) is either a cycle with short tails or a line with short tails and \f1(a) | = | yfa \ S 2 for each a eA. Consequently E(A,f) is an inverse semigroup, q.e.d. The requirement of the anti-inversibility of E(A,f) enforced a very simple structure of the unar (A,f). Proposition 5. Let (A,f) be a connected unar. E(A,f) is an anti-inverse semigroup if and only if(A,f) is a cycle of the cardinality I or 2 with at most one short tail. Proof. Suppose (A,f) has one of the required form. If E(A,f) is non-trivial, then either E(A,f) = {idA,f} or E(A,f) = {idA, f,f2}. Since E(A,f) is commutative, by [10] Theorem 4 (i) and (ii) it is anti-inverse. It is to be noted that as the multi-plicative table for E(A,f) \ {id^} = {/,/2} can serve the table 3) from [1] Example 2.1. Let (A,f) be a connected unar such that E(A,f) is an anti-inverse semigroup. Since E(A,f) is regular by [10] Theorem 1 or [1] Corollary 2.1 (i), we have in virtue of [12] Theorem 1 and [1] Theorem 2.1 (A,f) is a cycle of the cardinality at most 4 (except 3) with at most short tails. Admit \A° | J> 2. Assume a,beA°,a^ b. Since there exists geE(A,f) such that g(a) = b, g(6)eAV°\ we have g5 # g thus in regard with [1] Theorem 2.1 E(A,f) is not anti-inverse. Hence (A,f) is a cycle with at most one short tail. Then E(A,f) is a commutative monoid. Admitting | Am* | = 4, we have / 3 # / , which is a contradiction in virtue of [10] Theo-rem 4. Consequently | A™2 \ ;g 2, q.e.d. Remark. It is easy to verify that E(A,f) is anti-inverse for a connected unar (A,f) with | A | > 1 if and only if the groupoid (A, ef) has one of the following multiplicative table (or the other formed by a permutation of elements): ß/ a b a a a b a a ч a Ь a Ъ a Ъ Ъ a Ч a Ъ c a Ъ b b Ъ Ъ c Ъ c Ъ c Ъ 12 REFERENCES [I] Bogdanović, S., Milić, S. and Pavlović, V.: Anti-inverse semigroups, Publ. Inst. Math. Beograd, T. 24 (38), (1978), 19—28. [2] Čech, E.: Topological Spaces (гevised by Z. Frolík and M. Katëtov). Prague, Academia 1966. [3] Clifford, A. H., Preston, G. B.: Thз Algebгaic Theory of Semigroups, Amer. Math. Soc., Providence 1964, transl. Aлгeбpaичecкaя тeopия пoлyrpyкш, Mиp Mocквa 1972. [4] Chvalina, J.: Set transformations with centralizers formed by closed deformations of quasi-discrete topological spaces, Proc. Fourth Prague Top. Sym. 1976, Part B, Soc. Qвech. Math. Phys., Prague 1977, 83—89. [5] Chvalina, J.: Characterizations of certain monounary algebras I, Arch. Math. (Bmo) 14, 2 (1978), 85—98. [6] Chvalina, J.: Characterizations ofcertain monounary algebras II, ibid., 14, 3 (1978), 145—154. [7] Ježek, J., Kepka, T.: Semigroup representations of commutative idempotent Abelian groupoids, Comment. Math. Univ. Caгol. 16, 3 (1975), 487—500. [8] Kopeček, O.: Homomorphisms of machines I, Arch. Math. (Brno) 14, 1 (1978), 45—50. [9] Lorrain, F.: Topological spaces with minimum neighbourhoods, Amer. MatЬ. Monthly 76 (1969), 616—627. [10] Sharp, J. C, Jr.: Anti-regular semigroups, Publ. Inst. Math. Beograd, T. 24 (38), (1978), 147—150. [II] Skornjakov, L. A.: Unars, Colloquium on Universal Algebra, Esztergom 1977. To appear. [12] Skornjakov, L. A.: Unary algebras with regular endomorphism monoids, Acta Sci. Math. 40 (1978), 375—381. J. Chvalina 662 95 Brno, Janáčkovo nám. 2a Czechoslovakia 

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On connected unars with regular endomorphism monoids

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Institute of Mathematics AS CR, v. v. i.