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Algebraic structures within subsets of Hamel and Sierpinski-Zygmund functions.

1 Introduction

The symbols N, Q, and R denote the sets of positive integers, rational and real numbers, respectively. The cardinality of a set X is denoted by the symbol [absolute value of X]. In particular, [absolute value of N] is denoted by [omega] and [absolute value of R] is denoted by c. We consider only realvalued functions. No distinction is made between a function and its graph. For any two partial real functions f, g we write f + g, f - g for the sum and difference functions defined on dom(f) [intersection] dom(g). We write f|A for the restriction of f to the set A [subset or equal to] R. For any subset [GAMMA] of a vector space V over the field E, any v [member of] V, and any e [member of] E we define v + [GAMMA] = {v + y: y [member of] Y} and eY = {ey: y [member of] [GAMMA]}.

Recently, there have been lots of attention devoted to finding "large" structures (e.g., vector spaces, algebras) contained in various families of real functions (see [1, 3-6, 8-10, 12, 16, 18]). In this article we also consider "less restrictive" structures like groups and even semigroups. In case of many classes of functions the problem is trivially solved by using already known results about vector spaces contained in those classes (as these vector spaces have maximal possible dimensions). However, in certain situations looking for the "largest" group or semigroup may be of interest.

We will recall here some of the most recent definitions related to the theory of lineability (see [3,5, 6]). Let V be a vector space over the field E, F [subset or equal to] V, and k be a cardinal number. We say that F is star-like (with respect to E) if eF [subset or equal to] F for all e [member of] E\{0}. In addition, F is defined to be K-lineable (over E) if F [union] {0} contains a subspace of V of dimension [kappa]. The (coefficient of) lineability of the subset F over the field E is denoted by [L.sub.E](F) and defined as follows

[L.sub.E](F) = min{[kappa]: F is not [kappa]-lineable over E}.

In the case E = R we simply write L(F).

Proposition 1.1. Let V be a vector space over the field [E.sub.2] and [E.sub.1] be a subfield of [E.sub.2]. If F [subset or equal to] V is star-like with respect to [E.sub.2], then the following holds.

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) If [E.sub.1] is the smallest sub field of E2 and G is an additive group contained in F [union] {0}, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a vector subspace of V over [E.sub.1] contained in F [union] {0}.

Proof. (1) Choose any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and let W [subset or equal] V be a subspace over [E.sub.2] contained in F [union] {0} such that dim [E.sub.2](W) = [kappa]. Obviously, W is also a subspace when considered over [E.sub.1] and it can be verified that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Indeed, if {[f.sub.[??]]: [??] < [kappa]} is a basis of W over [E.sub.2], then {[q.sub.[lambda]][f.sub.[??]]: [??] < [kappa], [lambda] < dim[E.sub.1]([E.sub.2])} is a basis of W over E1, where {[q.sub.[lambda]]: [lambda] < dim[E.sub.1]([E.sub.2])} is a basis of [E.sub.2] over [E.sub.1].

Now, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then the largest possible [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and in this case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and consequently [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) Since G [subset or equal to] F [union] {0} and F is star-like obviously [E.sub.1]G [subset or equal to] F [union] {0}. Additionally, observe that

[E.sub.1] = {[+ or -][e.sub.n]/[e.sub.k]: k, [intersection] [member of] [Z.sub.+] and [e.sub.E] [not equal to] 0, where [e.sub.i] is the sum of i 1's} [union] {0}.

Therefore, for all [g.sub.1], [g.sub.2] [member of] G and [q.sub.1], [q.sub.2] [member of] [E.sub.1]\{0} we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, [E.sub.1]G is a vector subspace of V over [E.sub.1] contained in F [union] {0}.

Observe that in general, the weak inequality in part (1) cannot be replaced by equality neither strict inequality. Indeed, if F is a vector space over [E.sub.2], then there is equality in part (1). On the other hand if we pick V = [R.sup.R], [E.sub.1] = Q, [E.sub.2] = R, B to be a basis of V over [E.sub.2], and define F = [span.sub.Q](B) [union] [[union].sub.e[member of]R] eB then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence in that case there is strict inequality > in part (1).

As a consequence of the above proposition, let us note here that if [E.sub.1] = Q and [E.sub.2] = R then for any star-like F we have that [L.sub.Q](F) [greater than or equal to] [([dim.sub.Q](R)).sup.+] = [c.sup.+]. Additionally, every additive group contained in F U {0} has cardinality less than [L.sub.Q](F).

In this article we consider the following classes of functions. A function f: R [right arrow] R is:

* an extendability function provided there exists a connectivity function F: R x [0,1] [right arrow] R such that f(x) = F(x, 0) for every x [member of] R (f [member of] Ext);

* almost continuous (in sense of Stallings) if each open subset of [R.sup.2] containing the graph of f contains also the graph of a continuous function from R to R (f [member of] AC);

* Hamel function if the graph of f is a Hamel basis for [R.sup.2] (f [member of] HF);

* Sierpinski-Zygmund if for every set [GAMMA] [subset or equal to] R of cardinality continuum c, f|[GAMMA] is discontinuous (f [member of] SZ).

Recall here that the class of all continuous functions is contained in Ext, Ext [subset or equal to] AC, Ext [intersection] SZ = [empty set], AC [intersection] SZ [not equal to] [empty set] under additional set- theoretical assumptions (e.g., CH, Martin's Axiom), Ext [intersection] HF [not equal to] [empty set], AC [intersection] HF [not equal to] [empty set], and HF [intersection] SZ [not equal to] [empty set] (see [17]). In addition, a function f: R [right arrow] R is almost continuous if and only if it intersects every blocking set, i.e., a closed set K [subset or equal to] [R.sup.2] which meets every continuous function and is disjoint with at least one function from R to R. The domain of every blocking set contains a non-degenerate connected set. (See [11].) For f [member of] F [subset or equal to] [R.sup.R] we say that a set A [subset or equal to] R is f-negligible with respect to F if for every function g such that f[absolute value of (R\A) [equivalent to] g] (R\A) we have that g [member of] F.

It is known that L(SZ) > [c.sup.+] (see [9]) and that [2.sup.c]-lineability of SZ is undecidable in ZFC (see [10]). In [10] the authors also proved (Theorem 2.2) that for any c < [kappa] [less than or equal to] [2.sup.c], L(SZ) > [kappa] is equivalent to the existence of an additive group in SZ [union] {0} of cardinality [kappa]. This immediately implies the following property.

Remark 1.2. [L.sub.Q] (SZ) = L(SZ).

In the case of Hamel functions we have the following: L(HF) = 2 and [L.sub.Q] (HF) = [c.sup.+] (see [18]).

2 Semigroup in HF [intersection] SZ and lineability of AC [intersection] SZ

We will be using the following two lemmas to prove the existence of "large" semigroups in HF [intersection] SZ, Ext [intersection] HF, and AC [intersection] HF [intersection] SZ (under the assumption of CH).

Lemma 2.1. [17, Lemma 7] Let V [subset or equal to] [R.sup.n] be a Hamel basis and v' [member of] V. For each v [member of] V fix [q.sub.v] [member of] Q such that [q.sub.v'] [not equal to] -1. Then the set V' = {v + [q.sub.v]v': v [member of] V} is also a Hamel basis.

Lemma 2.2. There exists a function h [member of] HF [intersection] Ext and a set X [subset or equal to] R of cardinality c which is h-negligible with respect to Ext. Assuming CH, there exists a function h [member of] AC [intersection] HF [intersection] SZ and a set X [subset or equal to] R of cardinality c which is h- negligible with respect to AC.

Proof. Let F [subset or equal to] R be a linearly independent c-dense [F.sub.[sigma]] set (see [14, Theorem 11.7.2]). Then there exists a function f [member of] Ext such that R\F is f-negligible (see [7]). Using [17, Fact 6] we obtain the existence of a function h [member of] HF such that h|F [equivalent to] f |F. Obviously, h [member of] Ext and X = R\F is h-negligible with respect to Ext.

In the proof of Theorem 2 in [17] (page 123) a function h is constructed which belongs to AC [intersection] HF [intersection] SZ (under CH). One can easily see that this function h has a dense graph. It is known that for an almost continuous function f with a dense graph, every nowhere dense set is f-negligible with respect to AC (see [13]).

Theorem 2.3. Both HF [intersection] Ext and HF [intersection] SZ contain an additive semigroup of size [2.sup.c]. In addition, assuming CH, the same holds for AC [intersection] HF [intersection] SZ.

Proof. We will prove the statement for the family AC [intersection] HF [intersection] SZ. By the previous lemma, under the assumption of CH there exists a function h G AC [intersection] HF [intersection] SZ and a set X [subset or equal to] R of cardinality c which is h-negligible with respect to AC. Define H = {qh + h(0)g: q [member of] [Q.sub.+], g [member of] [Q.sub.+](X)} where [Q.sub.+] is the set of positive rationals and [Q.sub.+](X) = {f [member of] [R.sup.R]: f|(R\X) [equivalent to] 0 and f (x) [member of] [Q.sub.+] for x [member of] X}. Since h(0) [not equal to] 0 (for every f G HF, f (0) [not equal to] 0) we conclude that [absolute value of H] = [2.sup.c]. Next observe that [intersection] is closed under addition as both Q+ and Q+ (X) are closed under addition.

Finally we will justify that [intersection] [subset or equal to] AC [intersection] HF [intersection] SZ. Obviously [intersection] [subset or equal to] AC as AC is star-like and X is h-negligible with respect to AC. To see [intersection] [subset or equal to] HF recall that HF is star-like and then use Lemma 2.1 with V = h, v' = (0,/z(0)), and [q.sup.v] = g(x)/q for v = (x, h(x)), q [member of] [Q.sub.+], g [member of] [Q.sub.+](X) to conclude that h + h(0)g/q is a Hamel function (h + h{0)g/q is V' from Lemma 2.1). Consequently, qh + h(0)g = q(h + h(0)g/q) G HF.

To see [intersection] [subset or equal to] SZ recall that SZ is star-like and observe that h(0)g is a countably continuous function (e.g., union of countably many partial continuous functions) for all g [member of] [Q.sub.+](X). This implies qh + h(0)g [member of] SZ.

The existence of semigroups of cardinality [2.sup.c] in HF [intersection] Ext and HF [intersection] SZ can be justified in a very similar way (in the case of HF [intersection] SZ use X = R).

Theorem 2.4. Assume CH. Then L(AC [intersection] SZ) > [c.sup.+].

Proof. Let F = {[f.sub.[gamma]]: [gamma] < c} [subset or equal to] (AC [intersection] SZ) U {0} be a vector space of dimension < c. We will show that there exists an h [member of] Ac [intersection] SZ\F such that h + F [subset or equal to] AC [intersection] SZ. Since AC [intersection] SZ is star-like the latter will imply that {ah: a [member of] R} + F is a vector space in (AC [intersection] SZ) U {0} such that F [subset or equal to] {ah: a [member of] R} + F. Next using Zorn's lemma we will be able to conclude that (AC [intersection] SZ) U {0} contains a vector space of dimension [c.sup.+].

Let G = {[g.sub.[alpha]]: [alpha] < c} be the set of all continuous functions defined on [G.sub.[delta]] subsets of R = {[x.sub.[alpha]]: [alpha] < c}. For every [alpha] < c define [U.sub.[alpha]] to be the maximal open set such that dom([g.sub.[alpha]]\([[union].sub.[??]<[alpha]] [g.sub.[??]]) is residual in [U.sub.[alpha]]. We will construct by induction a sequence of partial functions [h.sub.[alpha]] ([alpha] < c) such that:

(i) [h.sub.[??]] [subset or equal to] [h.sub.[alpha]] for [subset or equal to] < [alpha];

(ii) [absolute value of dom([h.sub.[alpha]])] [less than or equal to] [omega] and [x.sub.[alpha]] [member of] dom([h.sub.[alpha]]);

(iii) ([g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]])) [subset or equal to] ([f.sub.[gamma]] + [h.sub.[??]]) for [zeta], [gamma] [less than or equal to] [??] [subset or equal to] < [alpha];

(iv) [f.sub.[gamma]] + [h.sub.[alpha]] is dense subset of ([g.sub.[zeta]]\[[union].sub.[xi]<[zeta]] [g.sub.[xi]])|[U.sub.[zeta]] for [zeta], [gamma] [less than or equal to] [alpha].

We start the construction of the sequence [h.sub.[alpha]]([alpha] < c) by defining [h.sub.0]([x.sub.0]) arbitrarily. Next choose a countable dense subset [D.sub.0] [subset or equal to] (dom([g.sub.0]) [intersection] [U.sub.0])\{[x.sub.0]} and put ([f.sub.0] + [h.sub.0])|[D.sub.0] [equivalent to] [g.sub.0]|[D.sub.0] (or equivalently [h.sub.0]|[D.sub.0] [equivalent to] ([g.sub.0] - [f.sub.0])|[D.sub.0]). It is easy to see that [h.sub.0] satisfies all the conditions (i)-(iv).

Now fix [alpha] < c and assume that the sequence [h.sub.[beta]] has been defined for all [beta] < [alpha] satisfying the conditions (i)-(iv). Put [h.sub.[alpha]] = [[union].sub.[beta]<[alpha]] [h.sub.[beta]]. If [x.sub.[alpha]] [not member of] dom([h.sub.[alpha]]), then choose

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next notice that since the conditions (i)-(iv) are satisfied for all [beta] < [alpha] to have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it suffices to assure the above condition for [zeta] = [alpha] or [gamma] = [alpha]. Choose a collection of pairwise disjoint countable sets [D.sub.[gamma],[xi]](([gamma] < [alpha] and [xi] = [alpha]) or ([gamma] = [alpha] and [zeta] [less than or equal to] [alpha])) contained in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

such that [D.sub.[gamma],[alpha]] is dense subset of dom([g.sub.[alpha]]\[[union].sub.[beta]<[alpha]][g.sub.[beta]]) [intersection] [U.sub.[alpha]] ([gamma] < [alpha]) and [D.sub.[alpha],[zeta]] is dense subset of dom([g.sub.[zeta]]\[[union].sub.[beta]<[zeta]][g.sub.[beta]]) [intersection] [U.sub.[zeta]] ([zeta] [less than or equal to] [alpha]). Note here that the above choice is possible as [absolute value of dom(([g.sub.[zeta]1] - [g.sub.[zeta]2]) [intersection] ([f.sub.[gamma]1] - [f.sub.[gamma]2]))] [less than or equal to] [omega] for [[zeta].sub.1], [[zeta].sub.2], [[gamma].sub.1], [[gamma].sub.2] [less than or equal to] [alpha], [[gamma].sub.1] [not equal to] [[gamma].sub.2] because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a continuous function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and we work under the assumption of CH. Now we define [f.sub.[gamma]] + [h.sub.[alpha]]|[D.sub.[gamma],[alpha]] [equivalent to] [g.sub.[alpha]]|] [D.sub.[gamma],[alpha]] for [gamma] < [alpha] and [f.sub.[alpha]] + [h.sub.[alpha]]|[D.sub.[alpha],[zeta]] [equivalent to] [g.sub.[zeta]]][D.sub.[alpha],[zeta]] for [zeta] [less than or equal to] [alpha]. This finishes the construction of ha. It is clear that [h.sub.[alpha]] satisfies the conditions (i), (ii), and (iv).

To see that the condition (iii) is also satisfied let us pick [subset or equal to] < [alpha] and [zeta], [gamma] [less than or equal to] [zeta]. By the inductive assumption we obtain that ([g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [[union].sub.[beta]<[alpha]] [h.sub.[beta]])) [subset or equal to] ([f.sup.[gamma]] + [h.sub.[zeta]]). Therefore to conclude that ([g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]])) [subset or equal to] ([f.sub.[gamma]] + [h.sub.[zeta]]) we need to justify that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The equality [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]]|{[x.sub.[alpha]]}) = [empty set] easily follows from the definition of [h.sub.[alpha]]([x.sub.[alpha]]). To see [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]]\[D.sub.[gamma]1,[alpha]]) = [empty set] ([[gamma].sub.1] < [alpha]) note that [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]]|[D.sub.[gamma]1,[alpha]]) = [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] - [f.sub.[gamma]1] + [g.sub.[alpha]])|[D.sub.[gamma]1,[alpha]]. If [gamma] = [[gamma].sub.1], then [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]]|[D.sub.[gamma]1,[alpha]]) = [g.sub.[zeta]] [intersection] [g.sub.[alpha]]|[D.sub.[gamma]1,[alpha]] = [empty set] as [D.sub.[gamma]1,[alpha]] [subset or equal to] dom([g.sub.[alpha]]\[[union].sub.[beta]<[alpha]][g.sub.[beta]]). If [gamma] [not equal to] [[gamma].sub.1]/ then [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]]|[D.sub.[gamma]1,[alpha]]) = ([g.sub.[zeta]] - [g.sub.[alpha]]) [intersection] ([f.sub.[gamma]] - [f.sub.[gamma]1])|[D.sub.[gamma]1,[alpha]] = [empty set]] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Very similarly we can justify that [g.sub.[zeta]] [intersection] ([f.sub.[gamma]] + [h.sub.[alpha]]|[D.sub.[alpha],[zeta]]) = [empty set] ([zeta] [less than or equal to] [alpha]). Hence the condition (iii) holds for [h.sub.[alpha]]. This finishes the inductive definition of the sequence [h.sub.[alpha]]([alpha] < c) satisfying the conditions (i)-(iv).

Define h = [[union].sub.[alpha]<c] [h.sub.[alpha]]. Obviously dom(h) = R. The conditions (ii)-(iii) imply that h + [f.sub.[gamma]] [member of] SZ for all [gamma] < c as any partial continuous function can be extended to a continuous function on a [G.sub.[delta]] subset of R (see [15]) and ([g.sub.[delta]] [intersection] ([f.sub.[gamma]] + h)) [subset or equal to] ([f.sub.[gamma]] + [h.sub.max([zeta],[gamma])]) for [zeta] < c.

Next we will argue that h + [f.sub.[gamma]] is almost continuous for every [gamma] < c. Let B [subset or equal to] [R.sup.2] be any blocking set. There exists a non-empty open interval I [subset or equal to] dom(B) and a continuous function g such that dom(g) is [G.sub.[delta]] dense subset of I and g [subset or equal to] B. Let [[zeta].sub.0] be the smallest ordinal number with the property that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is residual in I for some non-empty open interval I [subset or equal to] dom(B). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also residual in I (since we assume CH). Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and since [f.sub.[gamma]] + h is dense subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (condition (iv) for [alpha] = max([gamma], [[zeta].sub.0])) we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that h + [f.sub.[gamma]] [member of] AC.

Let us mention here that assuming GHC the above theorem implies that AC [intersection] SZ is [2.sup.c]-lineable and consequently L(AC [intersection] SZ) = L(SZ). On the other hand, there is a model of ZFC (see [2]) in which AC [intersection] SZ = [empty set]. These two observations imply the following.

Corollary 2.5.

(1) It is consistent with ZFC that L (AC [intersection] SZ) = L (SZ).

(2) It is consistent with ZFC that L(AC [intersection] SZ) < L(SZ).

It would be interesting to know if it is possible to have AC [intersection] SZ [not equal to] [empty set] and L(AC [intersection] SZ) < L(SZ). We state that as an open problem.

Problem 2.6. Is it consistent with ZFC that AC [intersection] SZ [not equal to] [empty set] and L(AC [intersection] SZ) < L(SZ)?

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[18] K. Plotka, On lineability and additivity of functions with finite preimages, J. Math. Anal. Appl. 421(2) (2015), 1396-1404.

Department of Mathematics, University of Scranton, Scranton, PA 18510, USA

email:Krzysztof.Plotka@scranton.edu

Received by the editors in December 2014.

Communicated by F. Bastin.

2010 Mathematics Subject Classification: Primary 15A03; Secondary 26A21, 03E75.
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