ThenAis said to be an algebra with easy direct limits

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Vol. 19 (2018), No. 1, pp. 291–302 DOI: 10.18514/MMN.2018.1744

MONOUNARY ALGEBRAS WITH EASY DIRECT LIMITS

EM´ILIA HALU ˇSKOV ´A Received 2015-09-21

Abstract. LetAbe an algebra such that exactly algebras isomorphic to a retract ofAcan be constructed fromAby direct limits. ThenAis said to be an algebra with easy direct limits. We will prove that ifAis a monounary, thenAis countable and the number of retracts ofAis not equal to@0. Further, we will see that the number of non-isomorphic monounary algebras with easy direct limits is2^@⁰.

2010Mathematics Subject Classification: 08B25; 08A60; 03E17; 03C99; 40A05; 40J05 Keywords: algebra, direct limit, retract, term, monounary algebra

1. INTRODUCTION

The importance of the notion of retract is well known and commonly appreciated in mathematics. There are many papers dealing with retracts of algebraic structures, see e.g. [7], [10] for non-complete review. Actual results concerning monounary algebras can be found in [1],[9] and [8].

The algebraic construction of direct limit is a well-known method of building up new algebras from given ones. It helps to make progress in particular topics, re- cently see e.g. [11], [12]. Furthermore it is interesting to investigate this construction generally, cf.[6], [15], [14].

LetAbe an algebra. We denote by L

!Athe class of all isomorphic copies of direct limits which can be obtained fromAand we denote byRAthe set of all retracts of A. ThenRA L

!A. We will say thatAisan algebra with easy direct limitsif every algebra from L

!Ais isomorphic to a retract ofA.

The aim of this paper is to investigate algebras with easy direct limits. IfA is finite, thenAis an algebra with easy direct limits, cf.[5]. Some infinite algebras with easy direct limits can be found in [4].

We will prove results concerning monounary algebras. Properties of monounary algebras with easy direct limits run from section 4. A monounary algebra with easy direct limits is countable. It contains a line or a cycle in every component. There

Supported by the grants VEGA 2/0028/13 and VEGA 2/0044/16.

c 2018 Miskolc University Press

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enter at most finitely many arrows to every point of it. The number of retracts of it is never equal to@⁰.

The section 5 contains a description of all monounary algebras with easy direct limits which have a bijective basic operation. This description implies that the number of non-isomorphic algebras with easy direct limits is equal to2^@⁰, see Thm.3.

2. PRELIMINARIES

The cardinality of a setA is denoted bykAk. The set of all positive integers is denoted byNand the set of all non-negative integers is denoted byN0.

We deal with monounary algebras. The fundamental operation is denoted byf. Short remarks about algebras of arbitrary type are before Lemma 6 and after Ex- ample1.

For monounary terminology see e.g. [10], [13].

LetAD.A; f /be a monounary algebra. A subsetB ofAis termed as achain ofAif for everya; b2Bthere isn2N0 eitherfⁿ.a/Db orfⁿ.b/Da. IfAis a chain ofA, then we will say thatAis abasic algebra.

Basic monounary algebras were introduced in [7] and will be used in the proof of Cor3.

If Ais connected and contains no cycle, then it is said to be a line, if for each a2Athere exists an uniqueb2Awithf .b/Da. A line is a basic algebra.

Denote

#bD fd 2AWf^k.d /Dbfor somek2N0g; f ¹.b/D fd 2AWf .d /Dbg;

"bD ff^k.b/Wk2N0g:

We remind the notion ofdegrees.a/of an elementa2A. Let us define byA^.¹^/ the set of all elements a2Asuch that the set #acontains an infinite chain of A.

Further, we putA^.0/D fa2AWf ¹.a/D¿g. Now we define a setA^./Afor each ordinal by induction. Assume that we have defined A^.˛/ for each ordinal

˛ < . Then we put

A^./D fa2An [

˛<

A^.˛/Wf ¹.a/ [

˛<

A^.˛/g:

The setsA^./ are pairwise disjoint. For eacha2A, eithera2A^.¹^/ or there is an ordinal witha2A^./. In the former case we puts.a/D 1, in the latter we set s.a/D.

LetBD.B; f /be a subalgebra ofA. ThenB is said to bea retract ofAif there exists an endomorphism h ofA such that h.A/DB andh².a/Dh.a/ for every a2A.

We will say thatB is abasic retract ofA, ifB is a retract ofAandB is a basic algebra.

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The notion of a direct limit we apply by [2],21.

LethP;ibe a directed partially ordered set. For eachp2P, letApD.Ap; f /be a monounary algebra. Assume that ifp; q2P; p¤q, thenAp\AqD¿. Suppose that for each pair of elementsp andq inP withp < q, we have a homomorphism 'pqofAp intoAqsuch thatp < q < simplies that'psD'pqı'qs:For eachp2P, suppose that'pp is the identity onAp. The familyfP;Ap; 'pqgis said to bedirect.

Assume thatp; q2P anda2Ap; b2Aq. Putab if there existss2P with ps; qssuch that'ps.a/D'qs.b/.

For eacha2 [

p2P

Ap putaD fb2 [

p2P

ApWabg:

DenoteAD faWa2 [

p2P

Apg:Definef .a/Df .a/:

Then the monounary algebra AD.A; f / is said to be a direct limit of the direct familyfP;Ap; 'pqg. We express this situation as follows

fP;Ap; 'pqg !A: (2.1)

Note that in the category theory this construction corresponds to (directed) colimit.

Let (1) be valid. The following lemmas follow from the definition of the direct limit.

Lemma 1. Letp2P anda; b2Ap. Ifa; bare in one component ofAp, thena; b are in one component ofA.

Lemma 2. Letp2P anda2Ap. Ifais a cyclic element ofAp, thenais a cyclic element ofA.

Lemma 3. Letu2Abelong to ann-element cycle ofA. Then there existp2P anda2Ap\usuch thatabelongs to ann-element cycle ofAp. Moreover, for all qpthe point'pq.a/belongs to ann-element cycle ofAq.

Lemma 4. Let u1; :::; u_k 2A. Then there existp2P anda1; :::; a_k 2Ap such thata12u1; :::; a_k2u_k.

Lemma 5. Let the operationf be injective (surjective) onApfor allp2P. Then the operationf is injective (surjective) onA.

LetAbe an algebra. The class L

!Ais created by all direct limits of families in which only algebras isomorphic toAoccur.

The following lemma will be repeatly used in this paper. It holds for algebras of arbitrary type and it is proved as Lemma 2 of [4]:

Lemma 6. IfA⁰2RAandB2 L

!A⁰, thenB2 L

!A.

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LetI be a nonempty set. For eachi2I letBi D.Bi; f /be a monounary algebra.

We denote byP

i2IBi a monounary algebra which is a disjoint union of algebras Bi,i2I.

LetADP

i2IBi andBi be connected for alli 2I. Leti2I. IfBi contains a cycle of lengthkfor somek2N, then we denote byCi a cycle of lengthk. Else we denote byCi a line. PutA^˘DP

i2ICi.

The following lemma is proved as Lemma 4 in [3].

Lemma 7. A^˘2 L

!A

Corollary 1. LetAD.A; f /be a monounary algebra. Then there existsB2 L

!A such that the operation ofBis bijective.

3. AUXILIARY RESULTS

In this section letAD.A; f /be a monounary algebra. We will prove five prelim- inary lemmas about algebras in L

!A.

Lemma 8. LetDD.D; f /be a component ofAD.A; f /and be an infinite cardinal number. If Ahas components isomorphic toD, then L

!A contains an algebra with more thancomponents isomorphic toD.

Proof. Letbe the cardinality ofD. Denote byB D.B; f /a subalgebra ofA which is created by components isomorphic to D. Take > andK the set of cardinality. Let i2K. Let Bi be a set of the cardinality , Bi\Bj D¿for i¤j. Letf be defined onBi in a such way that.Bi; f /ŠB.

TakeP the set of all finite subsets ofK. ConsiderhP;i. Letp2P. Put ApD f.d; p/Wd 2[

j2p

Bj[.AnB/g; f .d; p/D.f .d /; p/;

Ap D.Ap; f /:

We have.S

j2pBj; f /ŠB sincepis finite. ThereforeApŠA:

Let p; q 2P; p q and 'pq.x; p/D.x; q/ for every .x; p/2 Ap. Obviously fP;Ap; 'pqgis a direct family.

LetAD.A; f /be a direct limit of this family. ThenA2 L

!A. Ifi; j 2K; i¤j and di 2Bi; dj 2 Bj, then .di;fig/ and .dj;fjg/ are in isomorphic but different components of A. ThereforeA contains at least > components isomorphic to

D.

Lemma 9. Let be an infinite cardinal. Leta2AandDAbe such that (1) Df ¹.a/;

(2) .#d1[ "a; f /Š.#d2[ "a; f /for alld1; d22D;

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(3) Dis infinite.

Then there existsAD.A; f /2 L

!Asuch thatkAk> . Proof. DenoteBDS

d2D#d andD kBk. LetM be a set of cardinality greater than. Form2M letBmbe a set of cardinality ofandDmbe such thatDmBm

and kDmk D kDk. Assume that Bm\AD¿ and Bm\Bm⁰ D¿ for m¤m⁰. Further, letf be defined onBmin a such way thatf .t /Dafort2Dmand.Bm[ "

a; f /Š.B[ "a; f /.

LetP be the set of all finite subsets ofM. Letp2P. Put Ap D f.d; p/Wd 2 [

m2p

Bm[.AnB/g; f .d; p/D.f .d /; p/:

LetAp D.Ap; f /. Sincepis finite, we have thatApŠA:

Letp; q2P; pq. We define'pq.x; p/D.x; q/for every.x; p/2Ap. We have that'pq is a homomorphism fromApintoAq. ThusfP;Ap; 'pqgis a direct family.

LetAD.A; f /be a direct limit of this family.

Suppose thatv2A. Let.d; p/2v\Ap. Ifd2AnB, thenvD f.d; q/Wq2Pg. If d2Bmfor somem2p, thenvD f.d; q/Wq2 fs2P Wm2sgg D.d;fmg/. Further, .d;fmg/¤.t;fng/for everyd; t 2 [m2MDm,d ¤t andm; n2M.

Take

V D f.t;fmg/Wt 2Dm; m2Mg: We obtain

kAk kVk D X

m2M

kD_mk D kMk kDk kMk> :

Lemma 10. Let a2A be such that the set K D fd 2f ¹.a/Ws.d /D 1g is infinite. Then there existsAD.A; f /2 L

!Asuch thatkAk>kAk.

Proof. LetA⁰D.A⁰; f /2RAbe such thata2A⁰,f ¹.a/\A⁰DK andf is injective on#d for alld2K. The algebraA⁰satisfies assumptions of Lemma9for the elementaand the set K. TakeD kAk. Then there existsAD.A; f /2 L

!A⁰ such thatkAk>kAk. We obtainA2 L

!Aaccording to Lemma6.

Lemma 11. Letk2Nanda2Abe such that the setfd 2f ¹.a/Ws.d /kgis infinite. Then there existsAD.A; f /2 L

!Asuch thatkAk>kAk.

Proof. Denote LD fd 2f ¹.a/Ws.d /kg. The elements ofLhave degrees from the setf0; 1; :::; kg. The infiniteness ofLimplies that there existsi2 f0; 1; :::; kg such that the setLcontainskLkelements of degreei.

There existsA⁰D.A⁰; f /2RAsuch thata2A⁰,f ¹.a/\A⁰Df ¹.a/andf is injective on#d\A⁰for alld 2Lsuch thats.d /Di.

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The algebra A⁰ satisfies assumptions of Lemma9 for the elementa and the set DD fd2f ¹.a/Ws.d /Dig. TakeD kAk. Then there existsAD.A; f /2 L

!A⁰ such thatkAk>kAk. We obtainA2 L

!Aaccording to Lemma6.

Lemma 12. Leta2Aand the sequencefd_k; k2Ngbe such thatd_k 2f ¹.a/, s.dk/2Nands.dk/ < s.dkC1/for everyk2N. Then there existAD.A; f /2 L

!A andu2Asuch that the setfv2f ¹.u/Ws.v/D 1gis infinite

Proof. LetA⁰D.A⁰; f /2RAbe such thata2A⁰,f ¹.a/\A⁰Df ¹.a/and f is injective on#d_k\A⁰for allk2N. Put

DD [

k2N

.#d_k\A⁰/:

Leta⁰2D. Then fⁿ.a⁰/Dd_k for some uniquely determinedk2Nandn2N0, sincef is injective on#d_k\A⁰. We will writef ⁿ.d_k/Da⁰.

Let p2N. Put Ap D f.d; p/Wd 2A⁰g; f .d; p/D.f .d /; p/; Ap D.Ap; f /:

Let'p;pC1 be a mapping fromAp intoApC1 such that'p;pC1.d; p/D.d; pC1/

for d 2A⁰nD and'p;pC1.f ⁿ.d_k/; p/D.f ⁿ.d_k_C₁/; pC1/. This mapping is the homomorphism fromAp intoApC1. Consider a common linear order onN. We obtained a direct familyfN;Ap; 'pqg. LetAD.A; f /be a direct limit of this family.

TakeuD.a; 1/. ThenuD f.a; p/Wp2Ngand

f ¹.u/D f.di; 1/; .d1; i /Wi2Ng [ f.d; 1/Wd 2f ¹.a/n fdk; k2Ngg: Further, ifi2N, then

#.d1; i /D f.d1; i /; .f ¹.d1/; i /; :::; .f ^s.d¹^/.d1/; i /; .f ^s.d¹^{/ 1}.d2/; iC1/;

:::; .f ^s.d²^/.d2/; iC1/; .f ^s.d²^{/ 1}.d3/; iC2/; :::g

and f is injective on#.d1; i /. Thus s..d1; i //D 1. The set f.d1; i /W i 2Ng is infinite, because'_i;k.d₁; i /D.d₁_C_{k i}; k/.

4. THE MAIN RESULT

Several properties of monounary algebras with easy direct limits are proved in this section. They do not give a full description of all monounary algebras with easy direct limits but allow to observe some interesting cardinal characteristics, see Cor2 and Cor.3. Prop. 1can be generalized to universal algebras and this generalization is formulated in the end of this section. Other assertions are closely connected to an unary fundamental operation.

We will suppose thatAD.A; f /is an algebra with easy direct limits in proposi- tions 1 - 3.

Proposition 1. There existsBD.B; f /2RAsuch thatf is bijective onB.

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Proof. It follows from Cor.1.

Proposition 2. The number of components ofAwithout a cycle is finite and the number of components of A which contain an n-element cycle is finite for every n2N.

Proof. LetADP

k2KBk, whereBkare connected for allk2K.

Letn2N. Suppose thatAcontains infinitely manyn-element cycles. Denote K⁰D fk2KWB_k has a cycle of lengthng:

Letk2K. Consider an algebraDksuch thatDk DBk ifk…K⁰andDka cycle of lengthnifk2K⁰. TakeDDP

k2KD_k. We have D 2RA. There existsE 2 L

!D such that E contains more cycles of lengthnthanD in view of Lemma8. According to the number ofn-element cycles ofD we obtain thatE is not isomorphic to a subalgebra ofA. That meansE is not isomorphic to a retract ofA. In view of Lemma6we haveE2 L

!A.

Analogously we prove thatAis not an algebra with easy direct limits ifAcontains infinitely many components with a line.

Finally, suppose that A contains infinitely many components which contain no cycle and no line. In view of Lemma7there exists an algebra of L

!Asuch that it contains an infinitely many components with a line. ThereforeAis not an algebra

with easy direct limits.

Proposition 3. The setf ¹.a/is finite for everya2A.

Proof. We denoteK_aD fb2f ¹.a/Ws.b/D 1gfor everya2A.

Assume that there exists a2A such that Ka is infinite. We use Lemma 10 to obtain an algebraAD.A; f /2 L

!Awhich has the cardinality greater than the set A. ThusAis not isomorphic to a subalgebra ofAandAis not an algebra with easy direct limits, a contradiction.

Now suppose thata2Ais such that the setf ¹.a/is infinite. Thenfc2f ¹.a/W s.c/¤ 1gis infinite. The assumption of Lemma12fails to hold, otherwise there is A2 L

!Asuch thatAis not a retract ofA, which is a contradiction. Therefore there existsb2f ¹.a/withs.b/Dmaxfs.c/Wc2f ¹.a/Ws.c/¤ 1g. This implies that eithers.a/D 1or1 ¤s.a/Ds.b/C1.

Ifs.a/…N[ f1g, thens.b/!(where!is the least infinite ordinal number),b fulfils the assumption of Lemma12and this yields a contradiction too.

Further, lets.a/2N[ f1g. Thens.b/2Nandfc2f ¹.a/Ws.c/¤ 1g D fc2 f ¹.a/Ws.c/2Ng D fc 2f ¹.a/Ws.c/s.b/g. According to Lemma11again there existsA2 L

!Asuch thatAis not a retract ofA, which is a contradiction.

Theorem 1. IfAD.A; f /is an algebra with easy direct limits, then (i) every component ofAwithout a cycle contains a line,

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... ...

FIGURE1. The algebraA

(ii) the number of components ofAwithout a cycle is finite,

(iii) the number of components ofAwhich contain ann-element cycle is finite for everyn2N,

(iv) the setf ¹.a/is finite for everya2A.

Proof. Properties (ii) and (iii) are valid according to Prop.2. Therefore the number of lines of the algebraA^˘ is finite. This yields thatA^˘is isomorphic to a retract of Aaccording to Lemma7. Thus (i) is satisfied. Property (iv) is proved in Prop.3.

It is easy to see that conditions (i) - (iv) are independent.

Corollary 2. IfAD.A; f /is an algebra with easy direct limits, thenAis countable.

Proof. The number of components of Ais countable according to (ii) and (iii).

The number of elements of an arbitrary component ofAis countable according to

(iv).

Corollary 3. LetAD.A; f /be an algebra with easy direct limits. ThenkRAk ¤

@⁰.

Proof. We havekRAk 2^@⁰sinceAis countable.

Every component of A contains a cycle or a line in view of Thm. 1 (i). Thus every component ofAcontains a basic retract. Therefore there is no component of Awith the number of retracts equal to @⁰ according to Thm.5.9 of [7]. It implies kRAk ¤ @0according to Thm.5.10 of [7].

Example1. Let

AD f.0; 0/; .a; 0/; . a; 0/; .a; b/Wa; b2N; bag:

Letf .a; b/D.a; b 1/forb > 1 andf .a; 0/D.aC1; 0/. ThenAD.A; f / has properties (i)-(iv) and it is not an algebra with easy direct limits.

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... ...

...

... ... ... ... ... ... ...

FIGURE2. The algebraA

We hint whyA, see Fig.1, is not an algebra with easy direct limits. Considerd2A such that

1. s.d /D 1,

2. f is injective on#d,

3. there ise2An fdgsuch thatf .e/Df .d /.

The algebraAcontains exactly one pointd2Awhich satisfies the conditions 1. - 3, namelyd D.0; 0/.

Forn2NletAnD f.a; b; n/W.a; b/2Agandf .a; b; n/D.f .a; b/; n/. PutAnD .An; f /. Further, let'n;nC1.a; b; n/D.aC1; b; nC1/. Then fN;An; 'nmg is a direct family,AnŠA. LetfN;An; 'nmg !A, see Fig.2.

For everya > 0we have as on Fig.3

f ..a; a; nC1//Df .a; a; nC1/D.f .a; a/; nC1/D .a; a 1; nC1/D'n;nC1.a 1; a 1; n/D.a 1; a 1; n/:

Further,

.1; 1; n/¤.0; 0; n/andf ..1; 1; n//D.0; 0; n 1/Df ..0; 0; n//:

The elements .1; 1; n/ofAfor everyn2Nare pairwise different elements which satisfy conditions 1. - 3.

Remark1. Corollary1and Proposition1can be generalized for universal algebras by the following way:

LetAbe an algebra of typeF andf be an unary term operation overF such that f is an endomorphism of the algebraA. Then

(1) there existsBD.B; F /2 L

!Asuch thatf is bijective onB, (2) ifAis an algebra with easy direct limits, then

there existsBD.B; F /2RAsuch thatf is bijective onB.

5. ALGEBRAS WITH A BIJECTIVE OPERATION

LetAbe a nonempty set andf be a bijective operation fromAinto A. We will prove thatAD.A; f /is an algebra with easy direct limits if and only if the number of pairwise isomorphic components ofAis finite.

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...

(0,0,a)

(−a,0,1)=(0,0,a+1)

(0,0,a+2)

(4,4,a+5)

(3,3,a+4)

(2,2,a+3)

(1,1,a+2)

...

(1,1,2) (2,2,3) (3,3,4)

...

(a+4,4,5)

(a+3,3,4)

(a+2,2,3)

(a+1,1,2)

(a−1,0,1) (a+1,0,1)(a,0,1)

(4,4,5)

(−1,0,1) (0,0,1) (1,0,1)

...

FIGURE 3. A description ofA,a2N

Example2. LetAD.A; f /be monounary algebra such that every component of Ais a cycle of the length 2ⁿ for somen2Nand for everyn2Nthere is exactly one cycle of the length 2ⁿ. Then Ais an algebra with easy direct limts such that Ais countable andRAconsists of uncountable many non-isomorphic algebras. (A is an algebra with easy direct limits according to the fact that every algebra of L

!A contains a2-element cycle and by Lemmas2and3.)

The following assertion is obvious.

Lemma 13. LetAD.A; f /be a monounary algebra. The following conditions are equivalent:

(i) the operationf is bijective;

(ii)card f ¹.a/D1for everya2A;

(iii) ifBis a component ofA, thenBis a cycle or a line.

Theorem 2. LetAD.A; f /be a monounary algebra with a bijective operation.

Then the following conditions are equivalent:

(a) Ais an algebra with easy direct limits.

(b) The number of pairwise isomorphic components ofAis finite.

(c) Conditions(i) - (iv)from Thm.1 are satisfied.

Proof. The condition (a) implies (b) according to Lemma8.

Conditions (i) and (iv) from Thm.1 are satisfied sinceAhas a bijective operation.

Thus (b) is equivalent to (c).

Suppose that (b) is satisfied. Let (1) be valid andA_p ŠA. Then the operationf ofAis bijective by Lemma5.

LetB1; :::;B_k be pairwise isomorphic components ofA. Takeui an element of Bi fori D1; :::; k. According to Lemma4choose p2P anda1; :::; ak2Ap such

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thata12u1; :::; ak 2uk. Lemma1yields thata1; :::; akare in different components ofAp.

Let B1 be a cycle of length n. Then there exists q 2 P such that q p and fⁿ.'pq.ai//D'pq.ai/according to Lemma3 and the definition of a direct limit.

The element 'pq.ai/ belongs to an n element cycle, since 'pq.ai/Dui and ui

belongs to ann element cycle. In view of Lemma1elements'pq.a1/; :::; 'pq.a_k/ are in different components ofAq. Conclude thatAcontainskcycles of the length nas its components.

Now letB1be a line. Leti 2 f1; :::; kg. We get thatai is in no cycle according to the definition of a direct limit. Therefore the component ofAp, which possessesai, is a line, since the operation ofAp is bijective. Conclude thatAcontainsklines as its components.

We have proved thatAconsists of at most such number of isomorphic components asA.

LetAbe an algebra without a cycle. Then every component ofAis a line andA is a retract ofA.

Suppose thatAhas a cycle. Let us order lenghts of cycles of the algebraAac- cording to the divisibility and let minimal elements of this ordered set befki; i2Ig. We remark that the set I is nonempty and it is at most countable. Let i 2I. If a belongs to a ki-element cycle ofA, then every endomorphism ofAmapsainto a cycle of lengthki. Thusabelongs to a cycle of lengthki according to the direct limit definition. ThereforeAcontains a cycle of lengthki for everyi2I. That means that Ais a retract ofA.

Theorem 3. The number of all non-isomorphic monounary algebras with easy direct limits is equal to2^@⁰.

Proof. The number of all non-isomorphic monounary algebras with easy direct limits which have a bijective fundamental operation is equal to2^@⁰ according to the previous theorem. Therefore the assertion is valid in view of Cor.2.

Proposition 4. The number of monounary algebras with easy direct limits which have2^@⁰retracts is2^@⁰.

Proof. LetKN. Take an algebraAKsuch that (1) every component ofAK is a cycle,

(2) ifk2K, thenAKcontains exactly3cycles of lengthk, (3) ifk…K, thenA_Kcontains exactly2cycles of lengthk.

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REFERENCES

[1] Z. Farkasová and D. Jakub´ıková-Studenovská, “On radical classes of monounary algebras,”

Miskolc Math. Notes, vol. 14, no. 3, pp. 871–886, 2013.

[2] G. Gr¨atzer,Universal algebra, 2nd ed. Springer, New York, 2008. doi: 10.1007/978-0-387- 77487-9.

[3] E. Haluˇskov´a, “On iterated direct limits of a monounary algebra,” inContributions to general algebra, 10 (Klagenfurt, 1997). Heyn, Klagenfurt, 1998, pp. 189–195.

[4] E. Haluˇskov´a, “Two element direct limit classes of monounary algebras,”Math. Slovaca, vol. 52, no. 2, pp. 177–194, 2002.

[5] E. Haluˇskov´a and M. Ploˇsˇcica, “On direct limits of finite algebras,” inContributions to general algebra, 11 (Olomouc/Velk´e Karlovice, 1998). Heyn, Klagenfurt, 1999, pp. 101–103.

[6] J. Jakub´ık and G. Pringerov´a, “Direct limits of cyclically ordered groups,”Czechoslovak Math. J., vol. 44(119), no. 2, pp. 231–250, 1994.

[7] D. Jakub´ıková-Studenovská and J. Pócs, “Cardinality of retracts of monounary algebras,”

Czechoslovak Math. J., vol. 58(133), no. 2, pp. 469–479, 2008, doi:10.1007/s10587-008-0028-5.

[8] D. Jakub´ıková-Studenovská and J. Pócs, “On finite retract lattices of monounary algebras,”Math.

Slovaca, vol. 62, no. 2, pp. 187–200, 2012, doi:10.2478/s12175-012-0003-3.

[9] D. Jakub´ıková-Studenovská and J. Pócs, “Some properties of retract lattices of monounary algebras,”Math. Slovaca, vol. 62, no. 2, pp. 169–186, 2012, doi:10.2478/s12175-012-0002-4.

[10] Jakub´ıková-Studenovská, D. and Pócs, J.,Monounary algebras. P. J. ˇSafárik University in Koˇsice, 2009.

[11] F. Krajn´ık and P. Miroslav, “Compact intersection property and description of congruence lattices,”

Math. Slovaca, vol. 64, no. 3, pp. 643–664, 2014, doi:10.2478/s12175-014-0231-9.

[12] J. Lihov´a, “On convexities of lattices,”Publ. Math. Debrecen, vol. 72, no. 1-2, pp. 35–43, 2008.

[13] T. W. McKenzie R., McNulty G.,Algebras, Lattices, Varieties, vol.1. Wadsworth, 1987.

[14] C. Pelea, “On the direct limit of a direct system of multialgebras,”Discrete Mathematics, vol. 306, pp. 2916–2930, 2006.

[15] C. Pelea, “On convexities of lattices,”Publ. Math. Debrecen, vol. 72, no. 1-2, pp. 35–43, 2008.

Author’s address

Em´ılia Haluˇskov´a

Mathematical Institute, Slovak Academy of Sciences, Greˇs´akova 6, 040 01 Koˇsice, Slovakia E-mail address:ehaluska@saske.sk