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Analytic real algebras
SpringerPlus volume 5, Article number: 1684 (2016)
Abstract
In this paper we construct some real algebras by using elementary functions, and discuss some relations between several axioms and its related conditions for such functions. We obtain some conditions for realvalued functions to be a (edge) dalgebra.
Background
The notions of BCKalgebras and BCIalgebras were introduced by Iséki and Iséki and Tanaka (1980, 1978). The class of BCKalgebras is a proper subclass of the class of BCIalgebras. We refer useful textbooks for BCKalgebras and BCIalgebras (Lorgulescu 2008); Meng and Jun (1994); Yisheng (2006). The notion of dalgebras which is another useful generalization of BCKalgebras was introduced by Neggers and Kim (1999), and some relations between dalgebras and BCKalgebras as well as several other relations between dalgebras and oriented digraphs were investigated. Several aspects on dalgebras were studied (Allen et al. 2007; Han et al. 2010; Kim et al. 2012; Lee and Kim 1999; Neggers et al. 1999, 2000). Simply dalgebras can be obtained by deleting two identities as a generalization of BCKalgebras, but it gives more wide ranges of research areas in algebraic structures. Allen et al. (2007) developed a theory of companion dalgebras in sufficient detail to demonstrate considerable parallelism with the theory of BCKalgebras as well as obtaining a collection of results of a novel type. Han et al. (2010) defined several special varieties of dalgebras, such as strong dalgebras, (weakly) selective dalgebras and predalgebras, and they showed that the squared algebra \((X,\square , 0)\) of a predalgebra \((X, *, 0)\) is a strong dalgebra if and only if \((X,*, 0)\) is strong. Allen et al. (2011) introduced the notion of deformations in d / BCKalgebras. Using such deformations, dalgebras were constructed from BCKalgebras. Kim et al. (2012) studied properties of dunits in dalgebras, and they showed that the dunit is the greatest element in bounded BCKalgebras, and it is equivalent to the greatest element in bounded commutative BCKalgebras. They obtained several properties related with the notions of weakly associativity, dintegral domain, left injective in dalgebras also.
In this paper we construct some real algebras by using elementary functions, and discuss some relations between several axioms and its related conditions for such functions. We obtain some conditions for realvalued functions to be a (edge) dalgebra.
Preliminaries
A dalgebra (Neggers and Kim 1999) is a nonempty set X with a constant 0 and a binary operation "\(*\)" satisfying the following axioms:

(I)
\(x*x=0\),

(II)
\(0*x=0\),

(III)
\(x*y=0\) and \(y*x=0\) imply \(x=y\) for all \(x,y \in X\).
For brevity we also call X a dalgebra. In X we can define a binary relation "≤" by \(x \le y\) if and only if \(x*y=0\).
An algebra \((X,*, 0)\) of type (2,0) is said to be a strong dalgebra (Han et al. 2010) if it satisfies (I), (II) and (III\(^*\)) hold for all \(x, y\in X\), where

(III\(^*\)) \(x*y=y*x\) implies \(x=y\).
Obviously, every strong dalgebra is a dalgebra, but the converse need not be true (Han et al. 2010).
Example 1
(Han et al. 2010) Let \(\mathbf{R}\) be the set of all real numbers and \(e \in \mathbf{R}\). Define \(x*y :=(xy)\cdot (xe) + e\) for all \(x,y\in \mathbf{R}\) where "\(\cdot\)" and "−" are the ordinary product and subtraction of real numbers. Then \(x*x=e; e*x= e; x*y= y*x= e\) yields \((xy)\cdot (xe)=0\), \((yx)\cdot (ye)= 0\) and \(x=y\) or \(x=e=y\), i.e., \(x=y\), i.e., \((\mathbf{R},*, e)\) is a dalgebra.
However, \((\mathbf{R},*, e)\) is not a strong dalgebra. If \(x*y = y*x \Leftrightarrow (xy)\cdot (xe) + e = (yx)\cdot (ye) + e\) \(\Leftrightarrow (xy)\cdot (xe) = (xy)\cdot (ye) \Leftrightarrow (xy)\cdot (xe + ye) =0 \Leftrightarrow (xy)\cdot (x + y  2e) =0\) \(\Leftrightarrow (x=y\) or \(x+y = 2e\)), then there exist \(x=e+\alpha\) and \(y=e\alpha\) such that \(x+y= 2e\), i.e., \(x*y = y*x\) and \(x\not = y\). Hence, axiom (III\(^*\)) fails and thus the dalgebra \((\mathbf{R},*, e)\) is not a strong dalgebra.
A BCKalgebra is a dalgebra X satisfying the following additional axioms:

(IV)
\(((x*y)*(x*z))*(z*y)=0\),

(V)
\((x*(x*y))*y=0\) for all \(x,y,z \in X\).
Example 2
(Neggers et al. 1999) Let \(X:=\{0, 1, 2, 3, 4\}\) be a set with the following table:
*  0  1  2  3  4 

0  0  0  0  0  0 
1  1  0  1  0  1 
2  2  2  0  3  0 
3  3  3  2  0  3 
4  4  4  1  1  0 
Then \((X, *, 0)\) is a dalgebra which is not a BCKalgebra.
Let X be a dalgebra and \(x \in X\). X is said to be edge if for any \(x \in X\), \(x*X=\{x,0\}\). It is known that if X is an edge dalgebra, then \(x*0=x\) for any \(x \in X\) (Neggers et al. 1999).
Analytic real algebras
Let \(\mathbf{R}\) be the set of all real numbers and let “\(*\)” be a binary operation on \(\mathbf{R}\). Define a map \(\lambda : \mathbf{R}\times \mathbf{R}\rightarrow \mathbf{R}\). If we define \(x*y:=\lambda (x, y)\) for all \(x, y \in \mathbf{R}\), then we call such a groupoid \((\mathbf{R}, *)\) an analytic real algebra.
Given an analytic groupoid \((\mathbf{R}, *)\), we define
We call \(tr(*, \lambda )\) a trace of \(\lambda\). Note that the trace \(tr(*, \lambda )\) may or may not converge. Given an analytic groupoid \((\mathbf{R}, *)\), where \(x*y:= \lambda (x, y)\), if \(x*x = 0\) for all \(x\in \mathbf{R}\), then \(tr(*, \lambda ) =0\), but the converse need not be true in general.
Example 3
Let \(x_0 \in \mathbf{R}\). Define
Then \(tr(*, \lambda ) = \int ^{\infty }_{\infty } \,\,\, \lambda (x, x)\,\, dx = 0\), but \(\lambda (x_0, x_0)= 1\not = 0\), i.e., \(x_0 * x_0 \not = 0\).
Proposition 4
Let \((\mathbf{R}, *)\) be an analytic real algebra and let \(a, b, c\in \mathbf{R}\), where \(x*y:= ax + by + c\) for all \(x, y\in \mathbf{R}\). If \(tr(*, \lambda ) < \infty\), then \(tr(*, \lambda ) = 0\) and \(x*y = a(xy)\) for all \(x, y\in \mathbf{R}\).
Proof
Given \(x\in \mathbf{R}\), we have \(x*x = (a + b)x + c\). Since \(tr(*, \lambda ) < \infty\), we have \(\int ^{\infty }_{\infty } [(a + b) x + c] dx  < \infty\). Now \(\int ^{A}_0 [(a+ b)x + c] \,\,dx = (a+b)\frac{A^2}{2} + cA= A[\frac{a+b}{2}A + c]\) for a large number A, so that if \(tr(*, \lambda )<\infty\), then \(a+ b = 0\) and \(c=0\), i.e., we have \(x*y = a(xy)\), and thus \(x*x = 0\) for all \(x\in \mathbf{R}\). \(\square\)
Theorem 5
Let \(a, b, c, d, e, f \in \mathbf{R}\). Define a binary operation "\(*\)" on \(\mathbf{R}\) by
for all \(x, y\in \mathbf{R}\). If \(tr(*, \lambda )<\infty\) and \(0*x = 0\) for all \(x\in \mathbf{R}\), then \(x*y = ax(xy)\) for all \(x, y\in \mathbf{R}\).
Proof
Given \(x\in \mathbf{R}\), we have \(x*x = (a+ b + c)x^2 + (d+e)x + f\). Let \(A:= a + b + c\), \(B:= d + e\). If we assume \(tr(*, \lambda ) <\infty\), then \(\int ^{\infty }_{\infty } (Ax^2 + Bx + f) \,dx < \infty\). Now \(\int ^{L}_{0}(Ax^2 + Bx + f)\,\,dx = \frac{A}{3}L^3 + \frac{B}{2} L^2 + fL = L(\frac{A}{3} L^2 + \frac{B}{2} +f)\) for a large number L so that \(tr(*, \lambda ) <\infty\) implies \(A= B= f = 0\), i.e., \(a + b + c = 0, d+e = 0, f=0\). It follows that
If we assume \(0*x = 0\) for all \(x\in \mathbf{R}\), then, by (1), we have
for all \(x\in \mathbf{R}\). This shows that \(c=d = 0\). Hence \(x*y = ax(xy)\) for all \(x, y\in \mathbf{R}\). \(\square\)
Corollary 6
Let \(a, b, c, d, e, f \in \mathbf{R}\). Define a binary operation “\(*\)” on \(\mathbf{R}\) by
for all \(x, y\in \mathbf{R}\). If \(x*x = 0\) and \(0*x = 0\) for all \(x\in \mathbf{R}\), then \(x*y = ax(xy)\) for all \(x, y\in \mathbf{R}\).
Proof
The condition, \(x*x = 0\) for all \(x\in \mathbf{R}\), implies \(tr(*, \lambda ) <\infty\). The conclusion follows from Theorem 5. \(\square\)
Proposition 7
Let \(a, b, c, d, e, f \in \mathbf{R}\). Define a binary operation “\(*\)” on \(\mathbf{R}\) by
for all \(x, y\in \mathbf{R}\). If \(tr(*, \lambda )<\infty\) and the antisymmetry law holds for “\(*\)”, then \((ax cy + d)^2 + (aycx +d)^2 > 0\) for \(x\not = y\).
Proof
If \(tr(*, \lambda )<\infty\), then by (1) we obtain \(x*y = (ax cy + d)(xy)\). Assume the antisymmetry law holds for “\(*\)”. Then either \(x*y \not = 0\) or \(y*x \not = 0\) for \(x\not = y\). It follows that \((x*y)^2 >0\) or \((y*x)^2 > 0\), and hence \((x*y)^2 +(y*x)^2 > 0\). This shows that \((ax cy + d)^2 + (aycx +d)^2 > 0\). \(\square\)
Note that in Proposition 7 it is clear that if \((ax cy + d)^2 + (aycx +d)^2 > 0\) for \(x\not = y\), then the antisymmetry law holds.
Corollary 8
If we define \(x*y:= ax(xy)\) for all \(x, y\in \mathbf{R}\) where \(a\not = 0\), then \((\mathbf{R}, *)\) is a d algebra.
Proof
It is easy to see that \(x*x = 0 = 0*x\) for all \(x\in \mathbf{R}\). Assume that \(x\not = y\). Since \(x*y = ax(xy) = ax^2  axy\), by applying Proposition 7, we obtain \(b= a, c=0, d = e = f = 0\). It follows that \((ax  0y + 0)^2 + (ay  0x + 0)^2 = a^2x^2 + a^2y^2 = a^2(x^2 + y^2) > 0\) when \(a\not = 0\). By Proposition 7, \((\mathbf{R}, *)\) is a dalgebra. \(\square\)
Proposition 9
Let \(a, b, c, d, e, f \in \mathbf{R}\). Define a binary operation “\(*\)” on \(\mathbf{R}\) by
for all \(x, y\in \mathbf{R}\). If \(tr(*, \lambda )<\infty\) and \(x*0= x\) for all \(x\in \mathbf{R}\), then \(x*y = (1cy)(xy)\) for all \(x, y\in \mathbf{R}\).
Proof
If \(tr(*, \lambda )<\infty\), then by (1) we obtain \(x*y = (ax  cy + d)(xy)\) for all \(x, y\in \mathbf{R}\). If we let \(y:=0\), then \(x = x*0 = (ax + d)x\). It follows that \(ax^2 + (d1)x = 0\) for all \(x\in \mathbf{R}\). This shows that \(a=0, d=1\). Hence \(x*y = (1cy)(xy)\) for all \(x, y\in \mathbf{R}\). \(\square\)
Theorem 10
If we define \(x*y:= (ax  cy + d)(xy)\) for all \(x, y\in \mathbf{R}\) where \(a, c, d\in \mathbf{R}\) with \(a+ c\not = 0\), then the antisymmetry law holds.
Proof
Assume that there exist \(x\not = y\) in \(\mathbf{R}\) such that \(x*y = 0= y*x\). Then \((ax cy + d)(xy) = 0\) and \((ay  cx + d)(yx) = 0\). Since \(x\not = y\), we have
It follows that \((a+c)(xy)=0\). Since \(a+c \not = 0\), we obtain \(x=y\), a contradiction. \(\square\)
Remark
The analytic algebra \((\mathbf{R}, *)\), \(x*y= ax(xy)\) for all \(x, y\in \mathbf{R}\), was proved to be a dalgebra in Corollary 8 by using Proposition 7. Since \(x* y= ax(xy) = (ax  0y + 0)(xy)\), we know that \(a + 0 = a\not = 0\). Hence the algebra \((\mathbf{R}, *)\) can be proved by using Theorem 10 also.
Note that the analytic real algebra \((\mathbf{R}, *)\) discussed in Corollary 8 need not be an edge dalgebra, since \(x*0 = ax(x0)= ax^2 \not = x\).
Analytic real algebras with functions
Let \(\alpha , \beta : \mathbf{R} \rightarrow \mathbf{R}\) be realvalued functions. Define a binary operation “\(*\)” on \(\mathbf{R}\) by
where \(c\in \mathbf{R}\).
Proposition 11
Let \((\mathbf{R}, * )\) be an analytic real algebra defined by (3). If \(x * x = 0 = 0 * x\) for all \(x\in \mathbf{R}\), then \(x * y = 0\) for all \(x, y\in \mathbf{R}\).
Proof
Assume that \(x * x = 0\) for all \(x\in \mathbf{R}\). Then
If we let \(x:= 0\), then \(c=0\). If \(x\not = 0\), then \(\alpha (x) + \beta (x) = 0\), i.e., \(\beta (x) = \alpha (x)\) for all \(x\not = 0\) in \(\mathbf{R}\). It follows that
Assume \(0 * x = 0\) for all \(x\in \mathbf{R}\). Then
It follows that \(\beta (x) = 0\) for all \(x\not = 0\) in \(\mathbf{R}\). Hence we have \(x * y = 0\) for all \(x, y\in \mathbf{R}\). \(\square\)
Proposition 12
Let \((\mathbf{R}, * )\) be an analytic real algebra defined by (3). If \(x * x = 0\) and \(x * 0 = x\) for all \(x\in \mathbf{R}\), then \(x * y = xy\) for all \(x, y\in \mathbf{R}\).
Proof
If we assume \(x * x = 0\) for all \(x\in \mathbf{R}\), then by (4) we obtain \(x * y = \alpha (x)x  \alpha (y)y\). Assume that \(x * 0= x\) for all \(x\in \mathbf{R}\). Then \(x = x * 0 = \alpha (x)x  \alpha (0)0 = \alpha (x)x\). This shows that \(\alpha (x) = 1\) for any \(x\not = 0\) in \(\mathbf{R}\). Hence \(x * y = xy\) for all \(x, y\in \mathbf{R}\). \(\square\)
Let \(a, b_1, b_2, c, d, e: \mathbf{R} \rightarrow \mathbf{R}\) be realvalued functions and let \(f\in \mathbf{R}\). Define a binary operation “\(*\)” on \(\mathbf{R}\) by
for all \(x, y\in \mathbf{R}\). Assume \(0 * x = 0\) for all \(x\in \mathbf{R}\). Then
for all \(x\in \mathbf{R}\). It follows that \(f= 0\) and \(c(x)x + e(x)=0\) for all \(x\not = 0\) in \(\mathbf{R}\). Hence \(c(y)y^2 + e(y)y = 0\) for all \(y\in \mathbf{R}\). Hence
Assume \(x * x = 0\) for all \(x\in \mathbf{R}\). Then by (6) we obtain
It follows that \(d(x)x = [a(x)x^2 + b_1(x)b_2(x)x^2]\). By (6) we obtain
Theorem 13
Let \(b_1, b_2: \mathbf{R} \rightarrow \mathbf{R}\) be realvalued functions. Define a binary operation “\(*\)” on \(\mathbf{R}\) as in (7). If we assume \(b_2(x)x \not = b_2(y)y\) and \(b_1^2(x)x^2 + b_1^2(y)y^2 > 0\) for any \(x\not = y\) in \(\mathbf{R}\), then \((\mathbf{R}, * )\) is a d algebra.
Proof
Assume the antisymmetry law holds. Then it is equivalent to that if \(x\not = y\) then \(x * y \not = 0\) or \(y * x \not = 0\), i.e., if \(x\not = y\) then \((x * y)^2 + (y * x)^2 > 0\). Since \(x * y\) is defined by (7), we obtain that if \(x\not = y\) then
By assumption, we obtain that \((\mathbf{R}, *)\) is a dalgebra. \(\square\)
Example 14
Consider \(x * y : = ax(xy)\) for all \(x, y\in \mathbf{R}\). If we compare it with (7), then we have \(b_1(x) = a, b_2(y)= 1\) and \(b_2(x) = 1\) for all \(x\in \mathbf{R}\). This shows that \(b_2(x)x  b_2(y)y = (1)x  (1)y = yx \not = 0\) when \(x\not = y\). Moreover, \(b_1^2(x)x^2 + b_1^2(y)y^2 = a^2x^2 + b_1^2(y)y^2 >0\) since \(a\not = 0\). By applying Theorem 13, we see that an analytic real algebra \((\mathbf{R}, *)\) where \(x * y:= ax(xy)\), \(a \not = 0\) is a dalgebra.
Example 15
Consider \(x * y:= x\tan 2x [e^yy  e^x x]\) for all \(x, y\in \mathbf{R}\). By comparing it with (7), we obtain \(b_1(x) = \tan 2x, b_2(y)= e^y\) and \(b_2(x) = e^x\). If \(x\not = y\), then it is easy to see that \(xe^x \not = ye^y\) and \(b_1^2(x)x^2 + b_1^2(y)y^2 = (\tan 2x)^2x^2 + (\tan 2y)^2y^2 > 0\) when \(x\not = y\). Hence an analytic real algebra \((\mathbf{R}, * )\) where \(x * y:= x\tan 2x [e^yy  e^x x]\) is a dalgebra by Theorem 13.
In Theorem 13, we obtained some conditions for analytic real algebras to be dalgebras. In addition, we construct an edge dalgebra from Theorem 13 as follows.
Theorem 16
If we define a binary operation “\(*\)” on \(\mathbf{R}\) by
where \(b_1(x)\) is a realvalued function such that \(b_1(y) \not = 0\) if \(y\not = 0\). Then \((\mathbf{R}, *)\) is an edge d algebra.
Proof
Define a binary operation “\(*\)” on \(\mathbf{R}\) as in (7) with additional conditions: \(b_2(x)x \not = b_2(y)y\) and \(b_1^2(x)x^2 + b_1^2(y)y^2 > 0\) for any \(x\not = y\) in \(\mathbf{R}\). Assume \(x * 0 = x\) for all \(x\in \mathbf{R}\). Then
Combining with (7) we obtain
If we let \(xy\not = 0\), then
If we let \(x * y:= x\) when \(y=0\), then \((\mathbf{R}, *)\) is an edge dalgebra. \(\square\)
Example 17
Define a map \(b_1(x):= e^{\lambda x}\) for all \(x\in \mathbf{R}\). Then \(x * y = x[1\frac{e^{\lambda x}}{e^{\lambda y}}]= x(1e^{\lambda (xy)})\) when \(y\not = 0\). If we define a binary operation “\(*\)” on \(\mathbf{R}\) by
then \((\mathbf{R}, *)\) is an edge dalgebra.
Proposition 18
If we define a binary operation “\(*\)” on \(\mathbf{R}\) by
where \(b_1(x)\) is a realvalued function such that \(b_1(y) \not = 0\) if \(y\not = 0\). Assume that if \(x\not = y\), then either \(b_1(x * y) = b_1(x)\) or \(b_1(x * (x * y))= b_1(y)\). Then
for all \(x, y\in \mathbf{R}\).
Proof
By Theorem 16, \((\mathbf{R}, *)\) is an edge dalgebra and hence (8) holds for \(x * y =0\) or \(y=0\). Assume \(x * y\not = 0\) and \(y\not = 0\). Then
It follows that
proving the proposition. \(\square\)
Conclusions
We constructed some algebras on the set of real numbers by using elementary functions. The notions of (edge) dalgebras were developed from BCKalgebras, and widened the range of research areas. It is useful to find linear (quadratic) polynomial real algebras by using the real functions. In "Analytic real algebras" section, we obtained some linear (quadratic) algebras related to some algebraic axioms, and found suitable binary operations for (edge) dalgebras. In "Analytic real algebras with functions" section, we developed the idea of analytic methods, and obtained necessary conditions for the real valued function so that the real algebra is an edge dalgebra. We may apply the analytic method discussed here to several algebraic structures, and it may useful for find suitable conditions to construct several algebraic structures and many examples.
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Keywords
 Analytic real algebra
 Trace
 dalgebra
 BCKalgebra
Mathematics Subject Classification
 26A09
 06F35