#  Leibniz’s law and paraconsistent models of ZFC

Aldo Figallo-Orellano

###### Abstract

In this paper, we present full models for some Paraconsistent Set Theories
(PSTs). These models are built over Fidel semantics where they are specific
first-order structures in the sense of Model Theory. These structures are
known as $F$-structures in the literature and they are not algebras in the
universal algebra sense. We demonstrate how is possible to present
paraconsistent models for ZFC for any of PSTs studied in this paper, by
adapting the proofs given on the celebrated John Lane Bell’s books; in
general, we adapt the proofs in the mentioned book throughout the work.

##  1 Introduction

Paraconsistency is the study of logic systems having a negation $\neg$ which
is not explosive; that is, there exist formulas $\alpha$ and $\beta$ in the
language of the logic such that $\beta$ is not derivable from the
contradictory set $\\{\alpha,\neg\alpha\\}$. These systems are typically
called non-trivial systems and in this note we will work with this notion of
paraconsistency which is the most accepted for the logicians.

In the 1950’s, Nelson logic (even also called N3), was formulated by Nelson
and Markov as a result of certain philosophical objections to the
intuitionistic negation. The criticism concerned its disadvantageous non-
constructive property, namely that the derivability of the formula
$\neg(\alpha\wedge\beta)$ in an intuitionistic propositional calculus does not
imply that at least one of the formulas $\neg\alpha,\neg\beta$ is derivable. A
paraconsistent version of Nelson logic was first introduced in [1], where the
authors observed that the weaker system obtained from Nelson logic by deleting
the axiom schema $\alpha\to(\neg\alpha\to\beta)$ could be used to reason under
inconsistency without incurring in a trivial logic. Semantics for this
paraconsistent version of Nelson logic have been studied by Odintsov in [19],
who calls the logic N4, in terms of what is known in the literature as Fidel
structures and twist-structures.

On the other hand, the C-systems $C_{n}$ ($c<\omega$) and $C_{\omega}$ are
among the best-known contributions of da Costa and his collaborators. In
particular, for $C_{n}$ is possible to define a new operator $\circ$ where it
plays a role of recovering the classical theorems. These logics are known as
Logics of Formal Inconsistency, see for instance [10, 16]. Both $C_{n}$
($n<\omega$) and $C_{\omega}$ are non-algebrizable logics with Blok-Pigozzi
method, see [18]. This was one of the reasons why these systems remained
without semantics for several years. For every $C_{n}$, bi-valuation semantics
was first presented by da Costa and Alves exhibiting a soundness and
completeness Theorem, [9]. On the other hand, Fidel presented semantics for
both $C_{n}$ ($c<\omega$) and $C_{\omega}$ through introducing an algebraic-
relational structures that would bear his name; this idea was in fact based on
a generalization of Lindenbaum-Tarski process, [12]. Recently, Fidel semantics
for a first-order logic was presented in [13]; moreover, it was displayed the
relation with da Costa and Alves’s bi-valuation semantics through a
Representation Theorem. Other paraconsistent logics which are not
algebraizable were introduced recently by means of degree-preserving
construction in the paper [10, 11, 14].

On the other hand, since both $C_{n}$ and N4 can be axiomatized as an
extension of the intuitionistic positive caculus; we can present
$F$-structures for them based on Boolean and Heyting algebras.

Going to the central issue of this paper, recall that Boolean-valued models
were introduced with the intention to simply the Forcing method by Vopĕnka.
Afterwards, Scott, Solovay and Vopĕnka in 1965 proposed to present models for
Zermelo-Fraenkel Set Theory (ZF); an excellent exposition of the theory can be
found in the Bell’s book, [4]. This idea allowed obtaining models for
Intuitionistic version of Zermelo-Fraenkel Set Theory through working Heyting-
valued models, [5]. Using Boolean-valued models, it is possible to see that
the axiom of Choice is validated over them; i.e., it is commonly said that
Boolean-valued mode are models ZFC.

In this paper, we construct $F$-structures valued models as adapting the
machinery developed to Boolean and Heyting case. With these new structures, we
are able to prove that every axiom of ZF is validated over them; besides, we
discuss the central rôle of Leibniz law in the proof. These facts allow us to
consider paraconistent set theories over both (paraconsistent) Nelson’s and da
Costa’s logics, using the propositional axioms for first-order formulas and
set-theoretic axioms of Zermelo-Fraenkel. Moreover, we built bi-valuation
valued models and prove that they are good enough to get paraconsistent model
of ZFC.

##  2 Preliminaries

To present these logics, we first display some notation useful for the rest of
the paper. For a given logic $X$, let us consider the signature
$\Sigma=\\{\wedge,\vee,\to,\neg\\}$ for $X$, and the language $\mathfrak{Fm}$,
or set of formulas, over the denumerable set of variable $Var$. In the
following, we will consider several logics under the same signature $\Sigma$
and $X$ represents any of them. Recall now that the intuitionistic positive
caculus (${\bf Int^{+}}$) may be axiomatize by the following schemas and and
the rule Modus Ponens:

(Ax1)

$\alpha\mathbin{\rightarrow}(\beta\mathbin{\rightarrow}\alpha)$,

(Ax2)

$\big{(}\alpha\mathbin{\rightarrow}(\beta\mathbin{\rightarrow}\gamma)\big{)}\mathbin{\rightarrow}\big{(}(\alpha\mathbin{\rightarrow}\beta)\mathbin{\rightarrow}(\alpha\mathbin{\rightarrow}\gamma)\big{)}$,

(Ax3)

$(\alpha\land\beta)\mathbin{\rightarrow}\alpha$,

(Ax4)

$(\alpha\land\beta)\mathbin{\rightarrow}\beta$,

(Ax5)

$\alpha\mathbin{\rightarrow}\big{(}\beta\rightarrow(\alpha\land\beta)\big{)}$,

(Ax6)

$\alpha\mathbin{\rightarrow}(\alpha\lor\beta)$,

(Ax7)

$\beta\mathbin{\rightarrow}(\alpha\lor\beta)$,

(Ax8)

$(\alpha\mathbin{\rightarrow}\gamma)\mathbin{\rightarrow}\big{(}(\beta\mathbin{\rightarrow}\gamma)\mathbin{\rightarrow}(\alpha\lor\beta\mathbin{\rightarrow}\gamma)\big{)}$.

Paraconsistent Nelson’s logic, N4, is defined as an extension of ${\bf
Int^{+}}$ adding the following schemas: Axioms

(PN1)

$\sim(\sim\alpha)\leftrightarrow\,\alpha$,

(PN2)

$\sim(\alpha\vee\beta)\leftrightarrow\,\sim\alpha\,\wedge\sim\beta$,

(PN3)

$\sim(\alpha\wedge\beta)\leftrightarrow\,\sim\alpha\,\vee\sim\beta$,

(PN4)

$\sim(\alpha\to\beta)\leftrightarrow\alpha\,\wedge\sim\beta$.

To define the logic N3 , we only add the following axiom schema to N4

(Ax8)

$\alpha\to(\neg\alpha\to\beta)$

The logic $C_{\omega}$ is defined through adding the followings schemas to
${\bf Int^{+}}$

(C1)

$\alpha\lor\lnot\alpha$,

(C2)

$\lnot\lnot\alpha\mathbin{\rightarrow}\alpha$,

Now, the logic $C_{1}$ is defined by adding to the axioms set of $C_{\omega}$
and the following axiom schemas:

(C3)

$\alpha\vee(\alpha\rightarrow\beta)$,

(C4)

$\circ\alpha\rightarrow(\alpha\rightarrow(\neg\alpha\rightarrow\beta))$,

(C5)

$(\circ\alpha\wedge\circ\beta)\rightarrow(\circ(\alpha\wedge\beta)\wedge\circ(\alpha\vee\beta)\wedge\circ(\alpha\rightarrow\beta))$,
where $\circ\beta=\neg(\beta\wedge\neg\beta)$.

The notion of derivation of a formula $\alpha$ on any logic $X$ is defined as
usual. We say that $\alpha$ is derivable from $\Gamma$ in $X$, denoted by
$\Gamma\vdash\alpha$, if there exists a derivation of $\alpha$ from $\Gamma$
in $X$. If $\Gamma=\emptyset$ we denote $\vdash\alpha$; in this case, we say
that $\alpha$ is a theorem of $X$.

###  2.1 $F$-structures for N4

Now, recall that an algebra ${\bf A}=\langle A,\vee,\wedge,\to,0,1\rangle$ is
said to be a Heyting algebra if the reduct $\langle A,\vee,\wedge,0,1\rangle$
is a bounded distributive lattice and for any $a,b\in A$ the value of $a\to b$
is a pseudo-complement of $a$ with respect to $b$; i.e., the greatest element
of the set $\\{z\in A:a\wedge c\leq b\\}$. As a more general case, we say that
${\bf B}=\langle B,\vee,\wedge,\to,1\rangle$ is a generalized Heyting algebra
if $\langle B,\vee,\wedge,1\rangle$ is a distributive lattice with the
greatest element $1$ and $\to$ is defied as Heyting case. This last class of
algebras is called Relatively pseudo-complemented lattices in [20, Chapter IV]
and Implicative Lattices in [19].

Recall that a $F$-structure for N4 is a system $\langle{\bf
A},\\{N_{x}\\}_{x\in A}\rangle$ where $\bf A$ is a generalized Heyting algebra
and $\\{N_{x}\\}_{x\in A}$ is a family of sets of $A$ such that the following
conditions hold:

  * (i)

for any $x\in A$, $N_{x}\not=\emptyset$,

  * (ii)

for any $x,y\in A$, $x^{\prime}\in N_{x}$ and $y^{\prime}\in N_{y}$, the
following relations hold $x^{\prime}\vee y^{\prime}\in N_{x\wedge y}$ and
$x^{\prime}\wedge y^{\prime}\in N_{x\vee y}$, $x\in N_{x}^{\prime}$,

  * (iii)

for any $x,y\in A$, $y^{\prime}\in N_{x}$, we have $x\wedge y^{\prime}\in
N_{x\to y}$.

For the sake of simplicity, we will write sometimes $\langle{\bf A},N\rangle$
instead of $\langle{\bf A},\\{N_{x}\\}_{x\in A}\rangle$. Besides, we call the
$F$-structures for N4 by N4-structures. As example of N4-structure, we can
take a generalized Heyting algebra $\bf A$ and the set $N_{x}^{s}=A$ for every
$x\in A$. The structure $\langle A,\\{N_{x}^{s}\\}_{x\in A}\rangle$ will be
said to be a saturated one; these saturated structures are essential to prove
the Adequacy Theorem.

For a given set of formulas $\Gamma$ in N4, we will consider the binary
relation between formulas as follows:

$\alpha\equiv_{\Gamma}\beta$ iff
$\Gamma\vdash(\alpha\to\beta)\wedge(\beta\to\alpha).$

Bearing in mind the positive axioms of N4, (A1) to (A8), we have that
$\equiv_{\Gamma}$ is a congruence with respect to the connectives $\vee$,
$\wedge$ and $\to$. With $|\alpha|_{\Gamma}$ we denote the class of $\alpha$
under $\equiv_{\Gamma}$ and $\mathcal{L}_{\bf N4}$ denotes the set of all
classes. We can define the operations $\vee$, $\wedge$ and $\to$ on
$\mathcal{L}_{\bf N4}$ as follows:
$|\alpha\\#\beta|_{\Gamma}=|\alpha\\#\beta|_{\Gamma}$ with
$\\#\in\\{\wedge,\vee,\to\\}$. Thus, it is clear that $\langle\mathcal{L}_{\bf
N4},\wedge,\vee,\to,1\rangle$ is a generalized Heyting algebra with the
greatest element $1=|\alpha\to\alpha|_{\Gamma}$. To extend the latter algebra
to N4-structure, let us define the set $N_{|\alpha|}$ for each formula
$\alpha$ as follows:
$N_{|\alpha|_{\Gamma}}=\\{|\neg\beta|_{\Gamma}:\beta\equiv_{\Gamma}\alpha\\}$.
It is not hard to see that $\langle\mathcal{L}_{\bf
N4},\\{N_{|\alpha|_{\Gamma}}\\}_{\alpha\in\mathfrak{Fm}}\rangle$ is a
N4-structure that we will call Lindenbaum structure.

Besides, we say that a function $v:\mathfrak{Fm}\to\langle{\bf
A},\\{N_{x}\\}_{x\in A}\rangle$ is a N4-valuation if the following conditions
hold:

  * (v1)

$v(\alpha)\in A$ where $\alpha$ is an atomic formula,

  * (v2)

$v(\alpha\\#\beta)=v(\alpha)\\#v(\beta)$ where $\\#\in\\{\wedge,\vee,\to\\}$,

  * (v3)

$v(\neg(\alpha\vee\beta))=v(\sim\alpha)\wedge v(\sim\beta)$ and
$v(\neg(\alpha\wedge\beta)=v(\sim\alpha)\vee v(\sim\beta)$,

  * (v4)

$v(\neg(\alpha\to\beta))=v(\alpha)\wedge v(\sim\beta)$.

It is not hard to see that if we take a N4-valuation for atomic formulas, it
works well on formulas with more complexity as we can see in [19]. For us, a
formula $\alpha$ will be semantically valid from $\Gamma$ if for every
N4-structure and every valuation $v$ on the structure, the condition
$v(\beta)=1$ for every $\beta\in\Gamma$ implies $v(\alpha)=1$. In this case,
we denote $\Gamma\vDash\alpha$.

###### Theorem 2.1

[19, Theorem 3.8] Let $\Gamma\cup\\{\alpha\\}$ be a set of formulas of N4.
Then, $\Gamma\vdash\alpha$ if only if $\Gamma\vDash\alpha$

###  2.2 Bivaluational semantics for $C_{1}$

  * (bv1)

$v(\alpha\wedge\beta)=1$ iff $v(\alpha)=1$ and $v(\beta)=1$,

  * (bv2)

$v(\alpha\vee\beta)=1$ iff $v(\alpha)=1$ or $v(\beta)=1$,

  * (bv3)

$v(\alpha\to\beta)=1$ iff $v(\alpha)=0$ or $v(\beta)=1$,

  * (bv4)

$v(\alpha)=0$ implies $v(\neg\alpha)=1$,

  * (bv5)

$v(\neg\neg\alpha)=1$ implies $v(\alpha)=1$,

  * (bv5)

$v(\circ\alpha)=1$ implies $v(\alpha)=0$ or $v(\neg\alpha)=0$,

  * (bv7)

if $v(\circ(\alpha\wedge\beta))=0$ or $v(\circ(\alpha\vee\beta))=0$ or
$v(\circ(\alpha\rightarrow\beta))=0$, then $v(\alpha)=0$ or $v(\circ\alpha)=0$
or $v(\circ\beta)=0$.

We may define a semantics consequence relation $\vDash$ as follow:
$\Gamma\vDash\alpha$ iff for every valuation $v$, if $v(\Gamma)=\\{1\\}$, then
$v(\alpha)=1$. Thus, a strong version of soundness and completeness Theorem
for $C_{1}$ with respect to this semantics was proven in [9, Lemma 8]. It is
worth mentioning that it is not hard to see that, taking into account to
conditions (bv1) to( bv3), every bi-valuation $v:\mathfrak{Fm}\to\\{0,1\\}$
can be seen as homomorphism from $\mathfrak{Fm}$ into 2, where 2 is a two-
element Boolean algebra. Conversely, every homomorphism from $\mathfrak{Fm}$
into 2 that verifies (bv4) to (vb5) is a bi-valuation.

###  2.3 $F$-structures for $C_{1}$

It is possible to present other kind of semantics for $C_{1}$ through
$F$-structures, see, for instance, [18]. Recall that a $F$-structure for
$C_{1}$ ( $C_{1}$-structure) is a system $\langle{\bf A},\\{N_{x}\\}_{x\in
A},\\{O_{x}\\}_{x\in A}\rangle$ where $\bf A$ is a Boolean algebra and
$\\{N_{x},O_{x}\\}_{x\in A}$ is a family of sets of $A$ such that the
following conditions hold:

  * (i)

for very $x\in A$ there is $x^{\prime}\in N_{x}$ such that $x\vee
x^{\prime}=1$,

  * (ii)

for every $x^{\prime}\in N_{x}$ there is $x^{\prime\prime}\in N_{x^{\prime}}$
such that $x^{\prime\prime}\leq x$.

  * (iii)

for every $x\in A$ and every $y\in N_{x}$, there is $z\in O_{x}$ such that
$x\wedge y\wedge z=0$.

As on the case of N4, we write $\langle{\bf A},N,O\rangle$ instead of
$\langle{\bf A},\\{N_{x}\\}_{x\in A},\\{O_{x}\\}_{x\in A}\rangle$. We will
next consider the set of formulas $\mathcal{L}_{C_{1}}$ and taking again (A1)
to (A8), we have that $\equiv_{\Gamma}$ is a congruence with respect to the
connectives $\vee$, $\wedge$ and $\to$. With $|\alpha|_{\Gamma}$ where
$\equiv_{\Gamma}$ is defined as N4 case. Analogously to the case N4, we have
that
$\langle\mathcal{L}_{C_{1}},\\{N_{|\alpha|_{\Gamma}}\\}_{\alpha\in\mathfrak{Fm}}\rangle$
is a $C_{1}$-structure.

Besides, we say that a function $v:\mathfrak{Fm}\to\langle{\bf
A},\\{N_{x}\\}_{x\in A}\rangle$ is a $C_{1}$-valuation if the following
conditions hold:

  * (v1)

$v(\alpha)\in A$ where $\alpha$ is an atomic formula,

  * (v2)

$v(\alpha\\#\beta)=v(\alpha)\\#v(\beta)$ where $\\#\in\\{\wedge,\vee,\to\\}$,

  * (v3)

$v(\neg\alpha)\in N_{v(\alpha)}$, $v(\circ\alpha)\in O_{v(\alpha)}$ and
$v(\neg\neg\alpha)\leq v(\alpha)$,

  * (v4)

$v(\circ\alpha)\wedge v(\circ\beta)\leq v(\circ(\alpha\wedge\beta))\wedge
v(\circ(\alpha\vee\beta))\wedge v(\circ(\alpha\rightarrow\beta))$.

For us, a formula $\alpha$ will be semantically valid if every
$C_{1}$-structure $\langle{\bf A},N,O\rangle$ and every valuation $v$ on the
structure, then condition $v(\alpha)=1$ holds; and in this case, we denote
$\vDash\alpha$.

###### Theorem 2.2

[12, Theorem 5] Let $\alpha$ a formula of $C_{1}$. Then, $\vdash\alpha$ if
only if $\vDash\alpha$

Clearly, it is possible to prove a strong version of the last Theorem in a
similar way that was done in [18, Section 6] for non-algebraizable extensions
of $C_{\omega}$.

###  2.4 $F$-structures for $C_{\omega}$

Recall that a $F$-structure for $C_{\omega}$ ( $C_{\omega}$-structure) is a
system $\langle{\bf A},\\{N_{x}\\}_{x\in A}\rangle$ where $\bf A$ is a Boolean
algebra and $\\{N_{x}\\}_{x\in A}$ is a family of sets of $A$ such that the
following conditions hold:

  * (i)

for very $x\in A$ there is $x^{\prime}\in N_{x}$ such that $x\vee
x^{\prime}=1$,

  * (ii)

for every $x^{\prime}\in N_{x}$ there is $x^{\prime\prime}\in N_{x^{\prime}}$
such that $x^{\prime\prime}\leq x$.

Besides, we say that a function $v:\mathfrak{Fm}\to\langle{\bf
A},\\{N_{x}\\}_{x\in A}\rangle$ is a $C_{\omega}$-valuation if the following
conditions hold:

  * (v1)

$v(\alpha)\in A$ where $\alpha$ is an atomic formula,

  * (v2)

$v(\alpha\\#\beta)=v(\alpha)\\#v(\beta)$ where $\\#\in\\{\wedge,\vee,\to\\}$,

  * (v3)

$v(\neg\alpha)\in N_{v(\alpha)}$ and $v(\neg\neg\alpha)\leq v(\alpha)$.

A formula $\alpha$ is said to semantically consequence from $\Gamma$ if every
$C_{\omega}$-structure $\langle{\bf A},N\rangle$ and every valuation $v$ on
the structure, then condition $v(\alpha)=1$ holds; and in this case, we denote
$\vDash\alpha$. Moreover, we write $\Gamma\vDash\alpha$ if every
$C_{\omega}$-structure $\langle{\bf A},N\rangle$ and every valuation $v$ on
the structure, the condition $v(\beta)=1$ for every $\beta\in\Gamma$ implies
$v(\alpha)=1$.

###### Theorem 2.3

[18, Section 6] Let $\Gamma\cup\\{\alpha\\}$ be a set of formulas of
$C_{\omega}$. Then, $\Gamma\vdash\alpha$ if only if $\Gamma\vDash\alpha$

###  2.5 Zermelo-Fraenkel Set Theory

In this part of the paper, we will present Paraconsistent Set Theories build
over paraconsistent propositional logic expressing its axioms on a suitable
first-order version for Set Theory. We start by considering any logic $X$ over
the first-order signature ${\Theta}_{\omega}$ which contains an equality
predicate $\approx$ and a binary predicate $\in$. The system $ZF_{X}$ is the
first-order system obtained from the logic $X$ over $\Theta_{\omega}$ by
adding the following set-theoretic axiom schemas:

(Extensionality)

$\forall x\forall y[\forall z(z\in x\leftrightarrow z\in y)\to(z\approx y)]$,

(Pairing)

$\forall x\forall y\exists w\forall z[z\in w\leftrightarrow(z\approx x\vee
z\approx y)]$,

(Colletion)

$\forall x[(\forall y\in x\exists z\phi(y,z))\to\exists w\forall y\in x\exists
z\in w\phi(y,z)]$,

(Powerset)

$\forall x\exists w\forall z[z\in w\leftrightarrow\forall y\in z(y\in x)]$,

(Separation)

$\forall x\exists w\forall z[z\in w\leftrightarrow(z\in x\wedge\phi(z))]$,

(Empty set)

$\exists x\forall z[z\in x\leftrightarrow\neg(z\approx z)]$,

The set satisfying this axiom is, by extensionality, unique and we refer to it
with notation $\emptyset$.

(Union)

$\forall x\exists w\forall z[z\in w\leftrightarrow\exists y\in x(z\in y)]$,

(Infinity)

$\exists x[\emptyset\in x\wedge\forall y\in x(y^{+}\in x)]$,

From union and pairing and extensionality, we can note by $y^{+}$ the unique
set $y\cup\\{y\\}$.

(Induction)

$\forall x[(\forall y\in x\phi(y))\to\phi(x)]\to\forall x\phi(x)$.

These axioms are typically used to define Zermelo-Fraenkel Set Theory (for
short ZF), see [4]. In this paper, we will focus on the task of presenting
models for $ZF_{\bf N4}$, $ZF_{\bf N3}$, $ZF_{C_{1}}$ and $ZF_{C_{\omega}}$.

(Axiom of choice) (AC)

$\forall u\exists f[Fun(f)\wedge dom(f)\approx u\wedge\forall x\in
u[x\not\approx\emptyset\to f(x)\in x]]$.

Bell proved that (AC) implies Law of Excluded Middle (LEM) then (AC) does not
hold on Intuitionistic Zermelo Fraenkel set theory where (AC) is very weaker,
[3, 5]. The same occurs on $ZF_{\bf N4}$ ($ZF_{\bf N3}$ ) because N4 (N3) does
not hold LEM. In contrast, the systems $ZF_{C_{1}}$ and $ZF_{C_{\omega}}$ are
compatible on LEM.

##  3 Leibniz’s law and its models

In this section, we shall analyze the minimal conditions for models
constructed over a Heyting (or Boole) algebra that we need to prove that
several Zermelo-Fraenkel’s set-theoretic axioms are valid in these models.

Now, we fix a model of set theory $\mathbf{V}$ and a completed Heyting algebra
$A$. Let us construct a universe of names by transfinite recursion:

${\mathbf{V}_{\xi}}^{A}=\\{x:x\,\textrm{\rm a function and }\,ran(x)\subseteq
A\,\,\textrm{\rm and }\,dom(x)\subseteq\mathbf{V}_{\zeta}^{A}\,\textrm{\rm for
some}\,\zeta<\xi\\}$

${\mathbf{V}}^{A}=\\{x:x\in{\mathbf{V}_{\xi}}^{A}\,\textrm{\rm for some
}\xi\\}$

The class ${\mathbf{V}}^{A}$ is called the algebraic-valued model over $A$.
Let us observe that we only need the set $A$ in order to define
${\mathbf{V}_{\xi}}^{\langle A,N\rangle}$. By ${\cal L}_{\in}$, we denote the
first-order language of set theory which consists of only the propositional
connectives $\\{\to,\wedge,\vee\\}$ and two binary predicates $\in$ and
$\approx$. We can expand this language by adding all the elements of
${\mathbf{V}}^{\langle A,N\rangle}$; the expanded language we will denote
${\cal L}_{\langle A,N\rangle}$. For this construction of models we also have
Induction principles as the Boolean and Intuitionistic case, [4].

Now, we shall consider a minimal requirement for defining value of truth of
formulas in order to prove some of set-theoretic axioms of ZF are valid. Now,
for a given completed Heyting algebra $A$, the mapping $[\\![\cdot]\\!]^{\bf
A}:{\cal L}_{A}\to A$ is defined as follow:

$\displaystyle[\\![u\in v]\\!]^{\bf A}$ $\displaystyle=\bigvee\limits_{x\in
dom(v)}(v(x)\wedge[\\![x\approx u]\\!]^{A});$ (HV1)
$\displaystyle[\\![u\approx u]\\!]^{\bf A}$ $\displaystyle=1;$ (HV2)
$\displaystyle[\\![u\approx v]\\!]^{\bf A}$
$\displaystyle\leq[\\![\phi(u)\to\phi(v)]\\!]^{\bf A}\,\,\text{for all
formula}\,\,\phi;$ (HV3) $\displaystyle[\\![\varphi\\#\psi]\\!]^{\bf A}$
$\displaystyle=[\\![\varphi]\\!]^{\bf A}\\#[\\![\psi]\\!]^{\bf A}\,\,\text{for
every}\,\,\\#\in\\{\wedge,\vee,\to\\};$ (HV4) $\displaystyle[\\![\exists
x\varphi]\\!]^{\bf A}$ $\displaystyle=\bigvee\limits_{{u\in\mathbf{V}}^{\bf
A}}[\\![\varphi(u)]\\!]^{\bf A}\,\,\text{and}\,\,[\\![\forall
x\varphi]\\!]^{\bf A}=\bigwedge\limits_{{u\in\mathbf{V}}^{\bf
A}}[\\![\varphi(u)]\\!]^{\bf A}.$ (HV4)

$[\\![\varphi]\\!]$ is called the truth-value of the sentence $\varphi$ in the
language ${\cal L}_{A}$ in the algebraic-valued model over $A$. Besides, the
condition (HV3) is know as Leibniz’s law. This condition will be used in the
rest of the section.

###### Definition 3.1

A sentence $\varphi$ in the language ${\cal L}_{A}$ is said to be valid in
${\mathbf{V}}^{A}$, which is denoted by ${\mathbf{V}}^{\bf A}\vDash\varphi$,
if $[\\![\varphi]\\!]^{\bf A}=1$.

###### Lemma 3.2

For a given complete Heyting algebra $\bf A$. Then, $u,v\in{\mathbf{V}}^{\bf
A}$ we have

  * (i)

$[\\![u\approx v]\\!]^{\bf A}=[\\![v\approx u]\\!]^{\bf A}$,

  * (ii)

$u(x)\leq[\\![x\in u]\\!]^{\bf A}$ for every $x\in dom(u)$.

Proof.  (i) Let us consider the formula $\phi(z):=u=z$, then $[\\![u\approx
v]\\!]^{\bf A}\leq[\\![\phi(u)\to\phi(v)]\\!]^{\bf A}=[\\![u\approx
u]\\!]^{\bf A}\to[\\![v\approx u]\\!]^{\bf A}=1\to[\\![v\approx u]\\!]^{\bf
A}=[\\![v\approx u]\\!]^{\bf A}$. Analogously, we have $[\\![v\approx
u]\\!]^{\bf A}\leq[\\![u\approx v]\\!]^{\bf A}$.

(ii) $[\\![x\in u]\\!]^{\bf A}=\bigvee\limits_{z\in
dom(u)}(u(z)\wedge[\\![z\approx x]\\!]^{\bf A})\geq u(x)\wedge[\\![x\approx
x]\\!]^{\bf A}=u(x)$. $\square$

We will adopt the following notation, for every formula $\varphi(x)$ and every
$u\in\mathbf{V}^{\langle A,N\rangle}$: $\exists x\in u\varphi(x)=\exists
x(x\in u\wedge\varphi(x))$ and $\forall x\in u\varphi(x)=\forall x(x\in
u\to\varphi(x))$.

Now, we recall that for a given Heyting algebra $A$ we have the following
properties hold: (P1) $x\wedge y\leq z$ implies $x\leq y\Rightarrow z$ and
(P3) $y\leq z$ implies $z\Rightarrow x\leq y\Rightarrow x$ for any $x,y,z\in
A$.

Thus, we have the following

###### Lemma 3.3

Let $A$ be a complete Heyting algebra, for every formula $\varphi(x)$ and
every $u\in\mathbf{V}^{\bf A}$ we have

$[\\![\exists x\in u\varphi(x)]\\!]^{\bf A}=\bigvee\limits_{x\in
dom(u)}(u(x)\wedge[\\![\varphi(x)]\\!]^{\bf A}),$ $[\\![\forall x\in
u\varphi(x)]\\!]^{\bf A}=\bigwedge\limits_{x\in
dom(u)}(u(x)\to[\\![\varphi(x)]\\!]^{\bf A}).$

Proof.  Form the definition of $[\\![\cdot]\\!]^{\bf A}$ we have:

$[\\![\exists x\in u\varphi(x)]\\!]^{\bf A}=[\\![\exists x(x\in
u\wedge\varphi(x))]\\!]^{\bf A}=\bigvee\limits_{v\in\mathbf{V}^{\bf
A}}([\\![v\in u]\\!]^{\bf A}\wedge[\\![\varphi(v)]\\!]^{\bf
A})=\bigvee\limits_{v\in\mathbf{V}^{\bf A}}\bigvee\limits_{x\in
dom(u)}(u(x)\wedge[\\![x\approx v]\\!]^{\bf A}\wedge[\\![\varphi(v)]\\!]^{\bf
A})=\bigvee\limits_{x\in dom(u)}u(x)\wedge\bigvee\limits_{v\in\mathbf{V}^{\bf
A}}([\\![x\approx v]\\!]\wedge[\\![\varphi(v)]\\!]^{\bf A})$.

Now, we have $[\\![v\approx x\wedge\varphi(v)]\\!]^{\bf
A}\leq[\\![\varphi(x)]\\!]^{\bf A}$ and $[\\![x\approx
x\wedge\varphi(x)]\\!]^{\bf A}=[\\![\varphi(x)]\\!]^{\bf A}$. Therefore,

$\bigvee\limits_{z\in dom(u)}u(x)\wedge\bigvee\limits_{v\in\mathbf{V}^{\bf
A}}([\\![z\approx u]\\!]^{\bf A}\wedge[\\![\varphi(v)]\\!]^{\bf
A})=\bigvee\limits_{x\in dom(u)}[\\![u(x)\wedge\varphi(x)]\\!]^{\bf A}$.

On the other hand,

$[\\![\forall x\in u\varphi(x)]\\!]^{\bf A}=[\\![\forall x(x\in
u\to\varphi(x))]\\!]^{\bf A}=\bigwedge\limits_{v\in\mathbf{V}^{\bf
A}}[\\![v\in u]\\!]^{\bf A}\to[\\![\varphi(v)]\\!]^{\bf A}$

Then, we have

$\bigwedge\limits_{x\in dom(u)}[u(x)\to[\\![\varphi(x)]\\!]^{\bf
A}]\wedge[\\![v\in u]\\!]^{\bf A}=\bigwedge\limits_{x\in
dom(u)}[u(x)\to[\\![\varphi(x)]\\!]^{\bf A}]\wedge\bigvee\limits_{x\in
dom(u)}(u(x)\wedge[\\![v\approx x]\\!]^{\bf A})=\bigvee\limits_{x\in
dom(u)}(\bigwedge\limits_{x\in dom(u)}[u(x)\to[\\![\varphi(x)]\\!]^{\bf
A}]\wedge u(x)\wedge[\\![v\approx x]\\!]^{\bf A})\leq\bigvee\limits_{x\in
dom(u)}[\\![\varphi(x)]\\!]^{\bf A}\wedge[\\![v\approx x]\\!]^{\bf
A}\leq[\\![\varphi(v)]\\!]^{\bf A}$

Form the latter and (P1), we can conclude that $\bigwedge\limits_{x\in
dom(u)}[u(x)\to[\\![\varphi(x)]\\!]^{\bf A}]\leq[\\![v\in u]\\!]^{\bf
A}\to[\\![\varphi(v)]\\!]^{\bf A}$.

Now using Lemma 2.3 (ii) and (P3) we obtain

$\bigwedge\limits_{v\in\mathbf{V}^{\bf A}}[\\![v\in u]\\!]^{\bf
A}\to[\\![\varphi(v)]\\!]^{\bf A}\leq\bigwedge\limits_{v\in dom(u)}[\\![v\in
u]\\!]^{\bf A}\to[\\![\varphi(v)]\\!]^{\bf A}\leq\bigwedge\limits_{v\in
dom(u)}[\\![u(v)]\\!]^{\bf A}\to[\\![\varphi(v)]\\!]^{\bf A}$. $\square$

###### Definition 3.4

Let $A$ be a complete Heyting algebra. Given collection of sets $\\{u_{i}:i\in
I\\}\subseteq\mathbf{V}^{A}$ and $\\{a_{i}:i\in I\\}\subseteq A$, then mixture
$\Sigma_{i\in I}a_{i}\cdot u_{i}$ is the function $u$ with
$dom(u)=\bigcup\limits_{i\in I}dom(u_{i})$ and, for $x\in dom(u)$,
$u(x)=\bigvee\limits_{i\in I}a_{i}\wedge[\\![x\in u_{i}]\\!]^{\bf A}$.

The following result is known as Mixing Lemma and its proof is exactly the
same for intuitionistic case because it is an assertion about positive
formulas.

###### Lemma 3.5

[The Mixing Lemma] Let $u$ be the mixture $\Sigma_{i\in I}a_{i}\cdot u_{i}$.
If $a_{i}\wedge a_{j}\leq[\\![u_{i}\approx u_{j}]\\!]^{\bf A}$ for all $i,j\in
I$, then $a_{i}\leq[\\![u_{i}\approx u]\\!]^{\bf A}$.

A set $B$ refines a set $A$ if for all $b\in B$ there is some $a\in A$ such
that $b\leq a$. A Heyting algebra $H$ is refinable if every subset $A\subseteq
H$ there exists some anti-chain $B$ in $H$ that refines $A$ and verifies
$\bigvee A=\bigvee B$.

###### Theorem 3.6

Let $\bf A$ be a complete Heyting algebra such that $A$ is refinable. If
$\mathbf{V}^{\bf A}\vDash\exists x\psi(x)$, then there is $u\in\mathbf{V}^{A}$
such that $\mathbf{V}^{\bf A}\vDash\psi(u)$.

Proof.  Since $A$ is a set, then the collection $X=\\{[\\![\psi(u)]\\!]^{\bf
A}:u\in\mathbf{V}^{\bf A}\\}$ is also a set. By the Axiom of Choice, there is
an ordinal $\alpha$ and a set
$\\{u_{\zeta}:\zeta<\alpha\\}\subseteq\mathbf{V}^{\langle{\bf A},N\rangle}$
such that $X=\\{[\\![\psi(u_{\zeta})]\\!]^{\bf A}:\zeta<\alpha\\}$. Hence,
$[\\![\exists x\psi(x)]\\!]^{\bf
A}=\bigvee\limits_{\zeta<\alpha}[\\![\psi(u_{\zeta})]\\!]^{\bf A}$. Now let
$\\{a_{i}:i\in I\\}$ be a refinement of the set
$\\{[\\![\psi(u_{\zeta})]\\!]^{\bf A}:\zeta<\alpha\\}$ such that
$\bigvee\limits_{i\in
I}a_{i}=\bigvee\limits_{\zeta<\alpha}[\\![\psi(u_{\zeta})]\\!]^{\bf A}$. We
can choose for every $a_{i}$ some $v_{i}\in\mathbf{V}^{\langle{\bf
A},N\rangle}$ such that $a_{i}\leq[\\![\psi(v_{i})]\\!]^{\bf A}$. Now, let $u$
be the mixture $\Sigma_{i\in I}a_{i}\cdot v_{i}$. By the Mixing Lemma and
Leibniz’s law, we have $1=\bigvee\limits_{i\in I}a_{i}=\bigvee\limits_{i\in
I}\big{(}a_{i}\wedge[\\![v_{i}=u]\\!]^{\bf A}\big{)}\leq\bigvee\limits_{i\in
I}\big{(}[\\![\psi(v_{i})]\\!]^{\bf A}\wedge[\\![v_{i}=u]\\!]^{\bf
A}\big{)}\leq[\\![\psi(u)]\\!]^{\bf A}$ which completes the proof. $\square$

Now, given a complete Heyting $A^{\prime}$ of $A$, we have the associated
models $\mathbf{V}^{\bf A^{\prime}}$ and $\mathbf{V}^{\bf A}$. Then, it is
easy to see that $\mathbf{V}^{\bf A^{\prime}}\subseteq\mathbf{V}^{\bf A}$.

On the other hand, we say that a formula $\psi$ is restricted if all
quantifiers are of the form $\exists y\in x$ or $\forall y\in x$, then we have

###### Lemma 3.7

For any complete Heyting algebra $\bf A^{\prime}$ of $\bf A$ and any
restricted negation-free formula $\psi(x_{1},\cdots,x_{n})$ with variables in
$\mathbf{V}^{\bf A^{\prime}}$ the equality
$[\\![\psi(x_{1},\cdots,x_{n})]\\!]^{\bf
A^{\prime}}=[\\![\psi(x_{1},\cdots,x_{n})]\\!]^{\bf A}$.

Next, we are going to consider the Boolean algebra ${\bf
2}=(\\{0,1\\},\wedge,\vee,\neg,0,1)$ and the natural mapping
$\hat{\cdot}:\mathbf{V}^{\bf A}\to\mathbf{V}^{{\bf 2}}$ defined by
$\hat{u}=\\{\langle\hat{v},1\rangle:v\in u\\}$. This is well defined by
recursion on $v\in dom(u)$. Then, we have the following lemma holds:

###### Lemma 3.8

  * (i)

$[\\![u\in\hat{v}]\\!]^{\bf A}=\bigvee\limits_{x\in
v}[\\![u\approx\hat{x}]\\!]^{\bf A}$ for all $v\in\mathbf{V}$ and
$u\in\mathbf{V}^{\bf A}$,

  * (ii)

$u\in v\leftrightarrow\mathbf{V}^{\bf A}\vDash\hat{u}\in\hat{v}$ and
$u=v\leftrightarrow\mathbf{V}^{\bf A}\vDash\hat{u}=\hat{v}$,

  * (iii)

for all $x\in\mathbf{V}^{{\bf 2}}$ there exists a unique $v\in\mathbf{V}$ such
that $\mathbf{V}^{{\bf 2}}\vDash x=\hat{v}$,

  * (iv)

for any formula negation-free formula $\psi(x_{1},\cdots,x_{n})$ and any
$x_{1},\cdots,x_{n}\in\mathbf{V}$, we have
$\psi(x_{1},\cdots,x_{n})\leftrightarrow\mathbf{V}^{{\bf
2}}\vDash\psi(\hat{x_{1}},\cdots,\hat{x_{n}})$. Moreover for any restricted
negation-free formula $\phi$, we have
$\phi(x_{1},\cdots,x_{n})\leftrightarrow\mathbf{V}^{\bf
A}\vDash\phi(\hat{x_{1}},\cdots,\hat{x_{n}})$.

The proof of the last theorem is the same for intuitionistic case because we
consider restricted negation-free formulas and it will be used to prove the
validity of axiom Infinity.

###  3.1  Validating axioms

As for now, we are going to prove the validity of several set-theoretical
axioms of ZF. Therefore, let us consider a fix model $\mathbf{V}^{A}$. Then:

Pairing

Let $u,v\in\mathbf{V}^{\bf A}$ and consider the function $w=\\{\langle
u,1\rangle,\langle v,1\rangle\\}$. Thus, we have that $[\\![z\in w]\\!]^{\bf
A}=(w(u)\wedge[\\![z\approx u]\\!]^{\bf A})\vee(w(v)\wedge[\\![z\approx
v]\\!]^{\bf A})=[\\![z\approx u]\\!]^{\bf A}\vee[\\![z\approx v]\\!]^{\bf
A}=[\\![z\approx u\vee z\approx v]\\!]^{\bf A}$.

Union

Given $u\in\mathbf{V}^{\bf A}$ and consider that function $w$ with
$dom(w)=\bigcup\limits_{v\in dom(u)}dom(v)$ and $w(x)=\bigvee\limits_{v\in
A_{x}}v(x)$ where $A_{x}=\\{v\in dom(u):x\in dom(v)\\}$. Then,

$\displaystyle[\\![y\in w]\\!]^{\bf A}$ $\displaystyle=$
$\displaystyle\bigvee\limits_{x\in dom(w)}([\\![x\approx y]\\!]^{\bf
A}\wedge\bigvee\limits_{v\in A_{x}}v(x))$ $\displaystyle=$
$\displaystyle\bigvee\limits_{x\in dom(w)}\bigvee\limits_{v\in
A_{x}}([\\![x\approx y]\\!]^{\bf A}\wedge v(x))$ $\displaystyle=$
$\displaystyle\bigvee\limits_{v\in dom(u)}\bigvee\limits_{x\in
dom(v)}([\\![x\approx y]\\!]^{\bf A}\wedge v(x))$ $\displaystyle=$
$\displaystyle[\\![\exists v\in u(y\in v)]\\!]^{\bf A}.$

Separation

Given $u\in\mathbf{V}^{\bf A}$ and suppose $dom(w)=dom(u)$ and $w(x)=[\\![x\in
u]\\!]^{\bf A}\wedge[\\![\phi(x)]\\!]^{\bf A}$ then

$\displaystyle[\\![z\in w]\\!]^{\bf A}$ $\displaystyle=$
$\displaystyle\bigvee\limits_{x\in dom(w)}([\\![y\in w]\\!]^{\bf
A}\wedge[\\![\phi(y)]\\!]^{\bf A}\wedge[\\![y\approx z]\\!]^{\bf A})$
$\displaystyle\leq$ $\displaystyle\bigvee\limits_{x\in
dom(w)}([\\![\phi(z)]\\!]^{\bf A}\wedge[\\![y\approx z]\\!]^{\bf A}).$

Besides,

$\displaystyle[\\![\phi(z)]\\!]^{\bf A}\wedge[\\![y\approx z]\\!]^{\bf A}$
$\displaystyle=$ $\displaystyle\bigvee\limits_{y\in
dom(u)}(u(y)\wedge[\\![z\approx y]\\!]^{\bf A}\wedge[\\![\phi(z)]\\!]^{\bf
A})$ $\displaystyle\leq$ $\displaystyle\bigvee\limits_{y\in dom(u)}([\\![y\in
u]\\!]^{\bf A}\wedge[\\![z\approx y]\\!]^{\bf A}\wedge[\\![\phi(y)]\\!]^{\bf
A})$ $\displaystyle=$ $\displaystyle\bigvee\limits_{y\in
dom(u)}(w(y)\wedge[\\![z\approx y]\\!]^{\bf A})=[\\![z\in w]\\!]^{\bf A}.$

Infinity

Assume the formula $\psi(x)$ is $\emptyset\in x\wedge\forall y\in x(y^{+}\in
x)$. Then, the axiom in question is the sentence $\exists x\psi(x)$. Now, it
is clear that the negation-free formula $\emptyset\in x\wedge\forall y\in
x(y^{+}\in x)$ is restricted and certainly $\psi(\omega)$ is true. Hence, by
Lemma 3.8 (iv), we get $[\\![\psi(\hat{\omega})]\\!]^{\bf A}=1$, and so,
$[\\![\exists x\psi(x)]\\!]^{\bf A}=1$.

Collection

Given $u\in\mathbf{V}^{\bf A}$ and $x\in dom(u)$ there exists by Axiom of
Choice some ordinal $\alpha_{x}$ such that
$\bigvee\limits_{y\in\mathbf{V}^{\bf A}}[\\![\phi(x,y)]\\!]^{\bf
A}=\bigvee\limits_{y\in\mathbf{V}^{\bf
A}_{\alpha_{x}}}[\\![\phi(x,y)]\\!]^{\bf A}$. For $\alpha=\\{\alpha_{x}:x\in
dom(u)\\}$ and $v$ the function with domain $\mathbf{V}^{\bf A}$ and range
$\\{1\\}$, we have

$\displaystyle[\\![\forall x\in u\exists y\phi(x,y)]\\!]^{\bf A}$
$\displaystyle=$ $\displaystyle\bigwedge\limits_{x\in
dom(u)}(u(x)\to\bigvee\limits_{y\in\mathbf{V}^{\langle
A,N\rangle}}[\\![\phi(x,y)]\\!]^{\bf A})$ $\displaystyle=$
$\displaystyle\bigwedge\limits_{x\in
dom(u)}(u(x)\to\bigvee\limits_{y\in\mathbf{V}^{\langle
A,N\rangle}_{\alpha}}[\\![\phi(x,y)]\\!]^{\bf A})$ $\displaystyle=$
$\displaystyle\bigwedge\limits_{x\in dom(u)}(u(x)\to[\\![\exists y\in
v\phi(x,y)]\\!]^{\bf A})$ $\displaystyle=$ $\displaystyle[\\![\forall x\in
u\exists y\in v\phi(x,y)]\\!]^{\bf A}$ $\displaystyle\leq$
$\displaystyle[\\![\exists w\forall x\in u\exists y\in w\phi(x,y)]\\!]^{\bf
A}.$

##  4 Paraconsistent Nelson’s Set Theory

Now, we construct the class ${\mathbf{V}}^{\langle{\bf A},N\rangle}$ of
N4-structure-valued model over $\langle{\bf A},N\rangle$ following the
methodology developed for Heyting-valued models. By ${\cal L}_{\in}$, we
denote the first-order language of set theory which consists of the
propositional connectives $\\{\to,\wedge,\vee,\neg\\}$ of the N4 and two
binary predicates $\in$ and $\approx$. We can expand this language by adding
all the elements of ${\mathbf{V}}^{\langle{\bf A},N\rangle}$; the expanded
language we will denote ${\cal L}_{\langle{\bf A},N\rangle}$. Now, we will
define a valuation by induction on the complexity of a closed formula in
${\cal L}_{\langle{\bf A},N\rangle}$. Then, for a given complete and saturated
N4-structure $\langle{\bf A},N\rangle$, the mapping
$[\\![\cdot]\\!]^{\langle{\bf A},N\rangle}:{\cal L}_{\langle{\bf
A},N\rangle}\to\langle{\bf A},N\rangle$ is defined as follows:

$\displaystyle[\\![u\in v]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle=\bigvee\limits_{x\in dom(v)}(v(x)\wedge[\\![x\approx
u]\\!]^{\langle{\bf A},N\rangle});$ (N4-1) $\displaystyle[\\![u\approx
v]\\!]^{\langle{\bf A},N\rangle}$ $\displaystyle=\bigwedge\limits_{x\in
dom(u)}(u(x))\to[\\![x\in v]\\!]^{\langle{\bf
A},N\rangle})\wedge\bigwedge\limits_{x\in dom(v)}(v(x)\to[\\![x\in
u]\\!]^{\langle{\bf A},N\rangle});$ (N4-2)
$\displaystyle[\\![\varphi\\#\psi]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle=[\\![\varphi]\\!]^{\langle{\bf
A},N\rangle}\\#[\\![\psi]\\!]^{\langle{\bf A},N\rangle}\,\,\text{for
every}\,\,\\#\in\\{\wedge,\vee,\to\\};$ (N4-3)
$\displaystyle[\\![\neg\varphi]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle\in N_{[\\![\varphi]\\!]^{\langle{\bf
A},N\rangle}}\,\,\text{and}\,\,[\\![\neg\neg\varphi]\\!]^{\langle{\bf
A},N\rangle}=[\\![\alpha]\\!]^{\langle{\bf A},N\rangle};$ (N4-4)
$\displaystyle[\\![\neg(\varphi\vee\psi)]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle=[\\![\neg\varphi]\\!]^{\langle{\bf
A},N\rangle}\wedge[\\![\neg\psi]\\!]^{\langle{\bf
A},N\rangle}\,\,\text{and}\,\,[\\![\neg(\varphi\wedge\psi)]\\!]^{\langle{\bf
A},N\rangle}=[\\![\neg\varphi]\\!]^{\langle{\bf
A},N\rangle}\vee[\\![\neg\psi]\\!]^{\langle{\bf A},N\rangle};$ (N4-5)
$\displaystyle[\\![\neg(\varphi\to\psi)]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle=[\\![\varphi]\\!]^{\langle{\bf
A},N\rangle}\wedge[\\![\neg\psi]\\!]^{\langle{\bf A},N\rangle};$ (N4-6)
$\displaystyle[\\![\exists x\varphi]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle=\bigvee\limits_{{u\in{\mathbf{V}}^{\langle
A,N\rangle}}}[\\![\varphi(u)]\\!]^{\langle{\bf
A},N\rangle}\,\,\text{and}\,\,[\\![\forall x\varphi]\\!]^{\langle{\bf
A},N\rangle}=\bigwedge\limits_{{u\in{\mathbf{V}}^{\langle
A,N\rangle}}}[\\![\varphi(u)]\\!]^{\langle{\bf A},N\rangle};$ (N4-7)
$\displaystyle[\\![u\approx v]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle\leq[\\![\neg\phi(u)]\\!]^{\langle{\bf
A},N\rangle}\to[\\![\neg\phi(v)]\\!]^{\langle{\bf A},N\rangle}\,\,\text{for
any formula}\,\,\phi.$ (N4-8)

$[\\![\varphi]\\!]^{\langle{\bf A},N\rangle}$ is called the truth-value of the
sentence $\varphi$ in the language ${\cal L}_{\langle{\bf A},N\rangle}$ in the
$C_{\omega}$-structure-valued model over $\langle{\bf A},N\rangle$.

Now, we say that a sentence $\varphi$ in the language ${\cal L}_{\langle{\bf
A},N\rangle}$ is said to be valid in ${\mathbf{V}}^{\langle{\bf A},N\rangle}$,
which is denoted by ${\mathbf{V}}^{\langle{\bf A},N\rangle}\vDash\varphi$, if
$[\\![\varphi]\\!]^{\langle{\bf A},N\rangle}=1$.

For every completed N4-structure $\langle{\bf A},N\rangle$, the element
$\bigwedge\limits_{x\in A}x$ is the first element of $A$. So, ${\bf A}$ is a
complete Heyting algebra, we denote this element by ”$0$”. Besides, for every
closed formula $\phi$ of ${\cal L}_{\langle{\bf A},N\rangle}$ we have
$[\\![\phi]\\!]^{\langle{\bf A},N\rangle}\in A$ and so the proof of the
following Lemma can be given in the exact same way that was done in Lemmas 3.2

###### Lemma 4.1

For a given complete and saturated N4-structure $\langle{\bf A},N\rangle$.
Then, $[\\![u\approx u]\\!]^{\langle{\bf A},N\rangle}=1$, $u(x)\leq[\\![x\in
u]\\!]^{\langle{\bf A},N\rangle}$ for every $x\in dom(u)$, and $[\\![u\approx
v]\\!]^{\langle{\bf A},N\rangle}=[\\![v\approx u]\\!]^{\langle{\bf
A},N\rangle}$, for every $u,v\in{\mathbf{V}}^{\langle{\bf A},N\rangle}$.

The following fact can be checked by induction on the structure of formulas.

###### Lemma 4.2

For any complete and saturated N4-structure the following property holds
(Leiniz´s law): $[\\![u\approx v]\\!]^{\langle{\bf
A},N\rangle}\leq[\\![\phi(u)\to\phi(v)]\\!]^{\langle{\bf A},N\rangle}$ for any
formula $\phi$.

From the Lemmas 3.3 and 4.2, we have proven the following central result:

###### Lemma 4.3

Let $\langle{\bf A},N\rangle$ be a complete and saturated N4-structure. The,
for every formula $\varphi(x)$ and every $u\in\mathbf{V}^{\langle
A,N\rangle}$, we have:

$[\\![\exists x\in u\varphi(x)]\\!]^{\langle{\bf
A},N\rangle}=\bigvee\limits_{x\in
dom(u)}(u(x)\wedge[\\![\varphi(x)]\\!]^{\langle{\bf A},N\rangle}),$
$[\\![\forall x\in u\varphi(x)]\\!]^{\langle{\bf
A},N\rangle}=\bigwedge\limits_{x\in
dom(u)}(u(x)\to[\\![\varphi(x)]\\!]^{\langle{\bf A},N\rangle}).$

In the following Theorem, we will consider a complete and saturated
N4-structure $\langle{\bf A},N\rangle$, then $\bf A$ is indeed a complete
Heyting algebra. Moreover, for every formula $\psi$, we have that
$[\\![\psi]\\!]^{\langle{\bf A},N\rangle}$ belongs to $A$. From the latter and
taking into account the proofs displayed on Section 3.1, we have proven this
Theorem:

###### Theorem 4.4

Let $\langle{\bf A},N\rangle$ be a complete and saturated N4-structure. Then,
the set-theoretic axioms (Pairing), (Colletion), (Separation), (Empty set),
(Union), (Infinity) and (Induction) are valid in ${\mathbf{V}}^{\langle{\bf
A},N\rangle}$.

Now, we are in conditions of proving the axioms (Extensionality) and
(Powerset). Indeed,

###### Theorem 4.5

Let $\langle{\bf A},N\rangle$ be a complete and saturated N4-structure. Then,
the set-theoretic axioms (Extensionality), (Powerset) and (Empty set) are
valid in ${\mathbf{V}}^{\langle{\bf A},N\rangle}$.

Proof.  Given $x,y\in\mathbf{V}^{\langle{\bf A},N\rangle}$, then $[\\![\forall
z(z\in x\leftrightarrow z\in y)]\\!]^{\langle{\bf A},N\rangle}=$

$\displaystyle=$ $\displaystyle[\\![\forall z((z\in x\to z\in y)\wedge(z\in
y\to z\in x)]\\!]^{\langle{\bf A},N\rangle}$ $\displaystyle=$
$\displaystyle\bigwedge\limits_{z\in\mathbf{V}^{\langle A,N\rangle}}([\\![z\in
x]\\!]^{\langle{\bf A},N\rangle}\to[\\![z\in y]\\!]^{\langle{\bf
A},N\rangle})\wedge\bigwedge\limits_{z\in\mathbf{V}^{\langle
A,N\rangle}}([\\![z\in y]\\!]^{\langle{\bf A},N\rangle}\to[\\![z\in
x]\\!]^{\langle{\bf A},N\rangle})$ $\displaystyle\leq$
$\displaystyle\bigwedge\limits_{z\in dom(x)}([\\![z\in x]\\!]^{\langle{\bf
A},N\rangle}\to[\\![z\in y]\\!]^{\langle{\bf
A},N\rangle})\wedge\bigwedge\limits_{z\in dom(y)}([\\![z\in
y]\\!]^{\langle{\bf A},N\rangle}\to[\\![z\in x]\\!]^{\langle{\bf
A},N\rangle})$ $\displaystyle\leq$ $\displaystyle\bigwedge\limits_{z\in
dom(x)}(x(z)\to[\\![z\in y]\\!]^{\langle{\bf
A},N\rangle})\wedge\bigwedge\limits_{z\in dom(y)}(y(z)\to[\\![z\in
x]\\!]^{\langle{\bf A},N\rangle})$ $\displaystyle=$
$\displaystyle[\\![x\approx y]\\!].$

Thus, (Extensionality) is held.

Now, assume $u\in\mathbf{V}^{\langle{\bf A},N\rangle}$ and suppose $w$ a
function such that $dom(w)=\\{f:dom(u)\to A:f\,\hbox{function}\\}$ and
$w(x)=[\\![\forall y\in x(y\in u)]\\!]^{\langle{\bf A},N\rangle}$. Therefore,

$[\\![v\in w]\\!]^{\langle{\bf A},N\rangle}=\bigvee\limits_{x\in
dom(w)}([\\![\forall y\in x(y\in u)]\\!]^{\langle{\bf
A},N\rangle}\wedge[\\![x\approx v]\\!]^{\langle{\bf
A},N\rangle})\leq[\\![\forall y\in v(y\in u)]\\!]^{\langle{\bf A},N\rangle}.$

On the other hand, given $v\in\mathbf{V}^{\langle A,N\rangle}$ and consider
the function $a$ such that $dom(a)=dom(u)$ and $a(z)=[\\![z\in
u]\\!]^{\langle{\bf A},N\rangle}\wedge[\\![z\in v]\\!]^{\langle{\bf
A},N\rangle}$. So, it is clear that $a(z)\to[\\![z\in v]\\!]^{\langle{\bf
A},N\rangle}=1$ for every $z\in dom(a)$, therefore:

$\displaystyle[\\![\forall y\in v(y\in u)]\\!]^{\langle{\bf A},N\rangle}$
$\displaystyle=$ $\displaystyle\bigwedge\limits_{y\in dom(v)}(v(y)\to[\\![y\in
u]\\!]^{\langle{\bf A},N\rangle})$ $\displaystyle=$
$\displaystyle\bigwedge\limits_{y\in dom(v)}(v(y)\to([\\![y\in
u]\\!]^{\langle{\bf A},N\rangle}\wedge v(y)))$ $\displaystyle\leq$
$\displaystyle\bigwedge\limits_{y\in dom(v)}(v(y)\to a(y))$
$\displaystyle\leq$ $\displaystyle\bigwedge\limits_{y\in
dom(v)}(v(y)\to[\\![y\in a]\\!]^{\langle{\bf
A},N\rangle})\wedge\bigwedge\limits_{z\in dom(a)}(a(z)\to[\\![z\in
v]\\!]^{\langle{\bf A},N\rangle})$ $\displaystyle=$
$\displaystyle[\\![v\approx a]\\!]^{\langle{\bf A},N\rangle}.$

Since $a(y)\leq[\\![y\in u]\\!]^{\langle{\bf A},N\rangle}$ for every $y\in
dom(a)$ then we have $[\\![\forall y\in a(y\in u)]\\!]^{\langle{\bf
A},N\rangle}=1$. Now by construction we have that $a\in dom(w)$ and so,
$[\\![\forall y\in v(y\in u)]\\!]^{\langle{\bf A},N\rangle}\leq[\\![\forall
y\in a(y\in u)]\\!]^{\langle{\bf A},N\rangle}\wedge[\\![v\approx
a]\\!]^{\langle{\bf A},N\rangle}=w(a)\wedge[\\![v\approx a]\\!]^{\langle{\bf
A},N\rangle}\leq[\\![v\in w]\\!]^{\langle{\bf A},N\rangle}$. Therefore, the
axiom (Power set) holds.

Now, we show that (Empty set) is valid. Indeed, first let us note that
$[\\![u\approx u]\\!]^{\langle{\bf A},N\rangle}=1$ for all
$u\in\mathbf{V}^{\langle{\bf A},N\rangle}$ and then, $[\\![\neg(u\approx
u)]\\!]^{\langle{\bf A},N\rangle}\in N_{1}$. Therefore, let us consider a
function $w\in\mathbf{V}^{\langle{\bf A},N\rangle}$ such that $u\in dom(w)$
and $ran(w)\subseteq\\{[\\![\neg(u\approx u)]\\!]^{\langle{\bf
A},N\rangle}\\}$, then it is clear that $[\\![u\in w]\\!]^{\langle{\bf
A},N\rangle}=\bigvee\limits_{x\in dom(w)}(w(x)\wedge[\\![u\approx
x]\\!]^{\langle{\bf A},N\rangle})=[\\![\neg(u\approx u)]\\!]^{\langle{\bf
A},N\rangle}$ which completes the proof.

$\square$

Then we are in conditions to present the main result of this section:

###### Corollary 4.6

All set-theoretic axioms $ZF_{\bf N4}$ are valid in ${\mathbf{V}}^{\langle{\bf
A},N\rangle}$.

It is clear that we can be translated the proof on $ZF_{\bf N4}$ to the case
of $ZF_{\bf N3}$. So, we have the following Corollary:

###### Corollary 4.7

All set-theoretic axioms $ZF_{\bf N3}$ are valid in ${\mathbf{V}}^{\langle{\bf
A},N\rangle}$.

###  4.1  Leibniz’s law and Paraconsistent Set Theory $ZF_{\bf N4}$

In the next, we will present some remarks for the Paraconsistent Set Theory
$ZF_{\bf N4}$. From the condition (N4-8) allows us to prove Leibniz law and
this plays a fundamental role in the proof of Theorems 4.4 and 4.5, we will a
display an example where the law does not hold. Let us start by considering
the following three-element-chain Heyting algebra with a special negation,
$\mathbf{H}_{3}^{\neg}$, give by Table 1. Now, let us consider
${\mathbf{V}}^{\mathbf{H}_{3}^{\neg}}$ Heyting-like-valued model over
$\mathbf{H}_{3}^{\neg}$. Let $w\in{\mathbf{V}}^{\mathbf{H}_{3}^{\neg}}$ and
let $\psi(x)$ be the formula “$w\in x$” where $x$ is a free variable. Taking
$v=\\{\langle w,1\rangle\\}$, $u=\\{\langle w,\frac{1}{2}\rangle\\}$, we have
that $[\\![u\approx v]\\!]^{\mathbf{H}_{3}^{\neg}}=\frac{1}{2}$ and so
$[\\![u\approx
v]\\!]^{\mathbf{H}_{3}^{\neg}}\leq[\\![\psi(u)\to\psi(v)]\\!]^{\mathbf{H}_{3}^{\neg}}=1$
holds. But, since $[\\![\psi(v)]\\!]^{\mathbf{H}_{3}^{\neg}}=1$ and
$[\\![\psi(u)]\\!]^{\mathbf{H}_{3}^{\neg}}=\frac{1}{2}$, we have that
$[\\![\neg\psi(u)]\\!]^{\mathbf{H}_{3}^{\neg}}=\frac{1}{2}$ and
$[\\![\neg\psi(v)]\\!]^{\mathbf{H}_{3}^{\neg}}=0$, and therefore the Leibniz´s
law is not verified. This fact leads to the impossibility to prove the
validity of at least one axiom of ZF for formulas with negation; indeed, if we
take the formula $\phi(x):=\neg\psi(x)$, then it is not hard to see that the
axiom “Separation” is not valid. It is clear that the matrix
$\langle\mathbf{H}_{3}^{\neg},\\{1\\}\rangle$ validate all axioms of N4

$\rightarrow$ | 0 | $\bf\frac{1}{2}$ | 1
---|---|---|---
0 | $1$ | $1$ | $1$
$\bf\frac{1}{2}$ | $0$ | $1$ | $1$
1 | $0$ | $\frac{1}{2}$ | $1$

| $\neg$
---|---
0 | $1$
$\bf\frac{1}{2}$ | $\frac{1}{2}$
1 | $0$

Table 1: Tables of connectives $\rightarrow$ and $\neg$ in
$\mathbf{H}_{3}^{\neg}$

Now, we will present the study of the formula given above, but now using a
saturated N4-structure ${\cal
N}_{3}=\langle\\{0,\frac{1}{2},1\\},\wedge,\vee,\to,\neg
1,N_{0},N_{\frac{1}{2}},N_{1}\rangle$ where
$N_{0}=N_{\frac{1}{2}}=N_{1}=\\{0,\frac{1}{2},1\\}$ and $\to$ is given in
Table 1. Now, let’s now consider the model ${\mathbf{V}}^{{\cal N}_{3}}$ and
$w\in{\mathbf{V}}^{{\cal N}_{3}}$, and let $\psi(x)$ be the formula “$w\in x$”
where $x$ is a free variable. Then, we have the following possibilities:
$[\\![\neg\psi(v)]\\!]^{{\cal M}_{3}}=\frac{1}{2}$
and$[\\![\neg\psi(u)]\\!]^{{\cal N}_{3}}=1$, or $[\\![\neg\psi(v)]\\!]^{{\cal
N}_{3}}=\frac{1}{2}=[\\![\neg\psi(u)]\\!]^{{\cal N}_{3}}$. Clearly,
$[\\![\neg\neg\psi(u)]\\!]^{{\cal N}_{3}}=[\\![\psi(u)]\\!]^{{\cal N}_{3}}$
and $[\\![\neg\neg\psi(u)]\\!]^{{\cal N}_{3}}=[\\![\psi(u)]\\!]^{{\cal
N}_{3}}$ holds in N4. Thus, the model ${\mathbf{V}}^{{\cal N}_{3}}$ verifies
Leibniz’s law and it is a paraconsistent model. We do not know which is the
real value of the valuation of any formula with negation but we will know that
it is the right one.

Given a saturated complete N4-structure $\langle{\bf A},N\rangle$, for the
model ${\mathbf{V}}^{\langle{\bf A},N\rangle}$ and
$u,v\in{\mathbf{V}}^{\langle{\bf A},N\rangle}$, and a formula $\phi$, if
$[\\![\phi(u)]\\!]^{\langle{\bf A},N\rangle}=b$ and
$[\\![\phi(v)]\\!]^{\langle{\bf A},N\rangle}=a$, then we can always take
$[\\![\neg\phi(u)]\\!]^{\langle{\bf A},N\rangle}=a$ and
$[\\![\neg\phi(v)]\\!]^{\langle{\bf A},N\rangle}=b$. Clearly, the pair $(b,a)$
is an Leibniz one and it works well for axioms of N4; and what is more
important, ${\mathbf{V}}^{\langle{\bf A},N\rangle}$ continues being
paraconsistent.

##  5 Bi-valuation valued models for $C_{1}$ Paraconsistent Set Theory

Now, we construct the class ${\mathbf{V}}^{\bf 2}$ of bi-valuation valued
model following Section 3 but now using the Boolean algebra 2. By ${\cal
L}_{\in}$, we denote the first-order language of set theory which consists of
the propositional connectives $\\{\to,\wedge,\vee,\neg\\}$ of the $C_{1}$ and
two binary predicates $\in$ and $\approx$. Recall that $C_{1}$ has a non-
primitive connective $\circ$ and we want to consider formulas with this
connective on $ZF_{C_{1}}$. Let us now expand this language by adding all the
elements of ${\mathbf{V}}^{\bf 2}$; the expanded language we will denote
${\cal L}_{\bf 2}$. We will now define a valuation by induction on the
complexity of a closed formula in ${\cal L}_{\bf 2}$. Then, the mapping
$[\\![\cdot]\\!]^{\bf 2}:{\cal L}_{\bf 2}\to{\bf 2}$ is defined as follows:

$\displaystyle[\\![u\in v]\\!]^{\bf 2}$ $\displaystyle=\bigvee\limits_{x\in
dom(v)}(v(x)\wedge[\\![x\approx u]\\!]^{\bf 2});$ (BV1)
$\displaystyle[\\![u\approx v]\\!]^{\bf 2}$
$\displaystyle=\bigwedge\limits_{x\in dom(u)}(u(x))\to[\\![x\in v]\\!]^{\bf
2})\wedge\bigwedge\limits_{x\in dom(v)}(v(x)\to[\\![x\in u]\\!]^{\bf 2});$
(BV2) $\displaystyle[\\![\varphi\\#\psi]\\!]^{\bf 2}$
$\displaystyle=[\\![\varphi]\\!]^{\bf 2}\\#[\\![\psi]\\!]^{\bf 2},\text{for
every}\,\,\\#\in\\{\wedge,\vee,\to\\};$ (BV3)
$\displaystyle[\\![\varphi]\\!]^{\bf 2}$
$\displaystyle=0\,\,\text{implies}\,\,[\\![\neg\alpha]\\!]^{\bf 2}=1;$ (BV4)
$\displaystyle[\\![\neg\neg\varphi]\\!]^{\bf 2}$
$\displaystyle=1\,\,\text{implies}\,\,[\\![\varphi]\\!]^{\bf 2}=1;$ (BV5)
$\displaystyle[\\![\circ\varphi]\\!]^{\bf 2}$
$\displaystyle=1\,\,\text{implies}\,\,[\\![\varphi]\\!]^{\bf
2}=0\,\,\text{or}\,\,[\\![\neg\varphi]\\!]^{\bf 2}=0$ (BV6) $\text{The
condition}\,\,[\\![\circ(\varphi\wedge\psi)]\\!]^{\bf
2}=0\,\,\text{or}\,\,[\\![\circ(\varphi\vee\psi)]\\!]^{\bf
2}=0\,\,\text{or}\,\,[\\![\circ(\varphi\rightarrow\psi)]\\!]^{\bf
2}=0\,\,\text{implies:}\\\ $ $\displaystyle[\\![\varphi]\\!]^{\bf 2}$
$\displaystyle=0\,\,\text{or}\,\,[\\![\circ\varphi]\\!]^{\bf
2}=0\,\,\text{or}\,\,[\\![\circ\psi]\\!]^{\bf 2}=0;$ (BV7)
$\displaystyle[\\![u\approx v]\\!]^{\bf 2}$
$\displaystyle\leq[\\![\\#\phi(u)]\\!]^{\bf 2}\to[\\![\\#\phi(v)]\\!]^{\bf
2}\,\,\text{for any formula}\,\,\phi\,\,\text{and}\,\,\\#\in\\{\neg,\circ\\}.$
(BV8)

$[\\![\varphi]\\!]^{\bf 2}$ is called the truth-value of the sentence
$\varphi$ in the language ${\cal L}_{\bf 2}$ in the bi-valuation valued mode
over $\bf 2$. Now, we say that a sentence $\varphi$ in the language ${\cal
L}_{\bf 2}$ is said to be valid in ${\mathbf{V}}^{\bf 2}$, which is denoted by
${\mathbf{V}}^{\bf 2}\vDash\varphi$, if $[\\![\varphi]\\!]^{\bf 2}=1$.

The following Lemma will be used in the rest of section, an the proof is the
same to the Boolean case as we will see:

###### Lemma 5.1

The following properties true in ${\mathbf{V}}^{\bf 2}$:

  * (i)

$[\\![u\approx u]\\!]^{\bf 2}=1$,

  * (ii)

$u(x)\leq[\\![x\in u]\\!]^{\bf 2}$ for every $x\in dom(u)$,

  * (iii)

$[\\![u\approx v]\\!]^{\bf 2}=[\\![v\approx u]\\!]^{\bf 2}$, for every
$u,v\in{\mathbf{V}}^{\bf 2}$

  * (iv)

$[\\![u\approx v]\\!]^{\bf 2}\leq[\\![\phi(u)\to\phi(v)]\\!]^{\bf 2}$ for any
formula $\phi$ (Leibniz’s law).

Proof.  (i), (ii) and (iii): Bearing in mind that ${\mathbf{V}}^{\bf 2}$ is a
special Boolean-valued, the proof immediately follows from [4, Theorem 1.7]
item (i), (ii) and (iii), respectively. (iv): For negation free-formulas
$\psi$, applying [4, Theorem 1.7 (vii)], we have that $[\\![u\approx
v]\\!]^{\bf 2}\leq[\\![\psi(u)\to\psi(v)]\\!]^{\bf 2}$. Then, if $\psi$ is the
form $\neg\phi$ or $\circ\phi$, then applying (BV8) we have the condition (vi)
holds.

$\square$

###### Lemma 5.2

Let $\phi(x)$ be a formula such that ${\mathbf{V}}^{\bf 2}\vDash\exists
x\phi(x)$. Then,

  * (i)

For any $v\in{\mathbf{V}}^{\bf 2}$ there is $u\in{\mathbf{V}}^{\bf 2}$ such
that $[\\![\phi(u)]\\!]^{\bf 2}=1$ and $[\\![\phi(v)]\\!]^{\bf
2}=[\\![u\approx v]\\!]^{\bf 2}$.

  * (ii)

If $\psi(x)$ is a formula such that for any $u\in{\mathbf{V}}^{\bf 2}$, the
condition ${\mathbf{V}}^{\bf 2}\vDash\phi(u)$ implies ${\mathbf{V}}^{\bf
2}\vDash\psi(u)$, then ${\mathbf{V}}^{\bf 2}\vDash\exists
x[\phi(x)\to\psi(x)]$.

Proof.  (i): Applying Theorem 3.6 to the Boolean algebra $\bf 2$, we obtain
that there is $w\in{\mathbf{V}}^{\bf 2}$ such that $[\\![\phi(w)]\\!]^{\bf
2}=1$. Now, taking $b=[\\![\phi(v)]\\!]^{\bf 2}$ and the mixing function
$u=(b\cdot v)+(b^{*}\cdot w)$ where $b^{*}$ is the Boolean complement of $b$.
So, from Lemma 5.1 (iv) we have: $[\\![\phi(u)]\\!]^{\bf 2}\geq[\\![u\approx
v\wedge\phi(v)]\\!]^{\bf 2}\vee[\\![u\approx w\wedge\phi(w)]\\!]^{\bf 2}$. On
the other hand, applying Mixing Lemma (Lemma 3.5) for the set $\\{b,b^{*}\\}$,
we have $1=b\vee b^{*}\leq[\\![u\approx v\wedge\phi(v)]\\!]^{\bf
2}\vee[\\![u\approx w\wedge\phi(w)]\\!]^{\bf 2}$. Form the latter and (iv)
again, $[\\![u\approx v]\\!]^{\bf 2}=[\\![u\approx v]\\!]^{\bf
2}\wedge[\\![\phi(u)]\\!]^{\bf 2}\leq[\\![\phi(v)]\\!]^{\bf 2}$. Since
$b=[\\![\phi(v)]\\!]^{\bf 2}\leq[\\![u\approx v]\\!]^{\bf 2}$ which completes
the proof.

(ii): Let $u\in{\mathbf{V}}^{\bf 2}$. From (i), there exists
$u\in{\mathbf{V}}^{\bf 2}$ such that $[\\![\phi(u)]\\!]^{\bf 2}=1$ and
$[\\![\phi(v)]\\!]^{\bf 2}=[\\![u\approx v]\\!]^{\bf 2}$. Thus,
$[\\![\phi(v)]\\!]^{\bf 2}\leq[\\![\phi(v)]\\!]^{\bf 2}$ as desired. $\square$

Next, we will define the notion of core of a element of ${\mathbf{V}}^{\bf
2}$.

###### Definition 5.3

Let $u\in{\mathbf{V}}^{\bf 2}$, a set $v\subseteq{\mathbf{V}}^{\bf 2}$ is call
core for $u$ if the following holds: (i) $[\\![x\in u]\\!]^{\bf 2}=1$ for all
$x\in v$ and (ii) for each $y\in{\mathbf{V}}^{\bf 2}$ such that $[\\![y\in
u]\\!]^{\bf 2}=1$ there is a “unique” $x\in v$ such that $[\\![x\approx
y]\\!]^{\bf 2}=1$.

###### Lemma 5.4

Any $u\in{\mathbf{V}}^{\bf 2}$ has a core.

Proof.  Let us consider $a_{x}=\\{\langle z,u(z)\wedge[\\![z\approx
x]\\!]^{\bf 2}\rangle:z\in dom(u)\\}$ for every $x\in{\mathbf{V}}^{\bf 2}$. By
Collection on the set $\\{f\in{\bf 2}^{dom(u)}:\,\,\text{\em there
is}\,\,z\in{\mathbf{V}}^{\bf 2}\,\,\text{\em such that}\,\,f=a_{y}\\}$, there
is $w\subseteq{\mathbf{V}}^{\bf 2}$ such that for all $y\in{\mathbf{V}}^{\bf
2}$ there exists $x\in w$ such thar $a_{x}=a_{y}$. Let now $C=\\{x\in
w:[\\![x\in u]\\!]^{\bf 2}=1\\}$ be a set, for any $y$ if $x$ is such that
$a_{x}=a_{y}$ then $u(z)\wedge[\\![z\approx x]\\!]^{\bf
2}=u(z)\wedge[\\![z\approx y]\\!]^{\bf 2}$ for all $z\in dom(u)$. Therefore,
$[\\![y\in u]\\!]^{\bf 2}=1$ and then $x\in w$ such that
$1=\bigvee\limits_{z\in dom(u)}[u(z)\wedge[\\![z\approx y]\\!]^{\bf
2}]=\bigvee\limits_{z\in dom(u)}[u(z)\wedge[\\![z\approx y]\\!]^{\bf
2}\wedge[\\![x\approx y]\\!]^{\bf 2}]\leq[\\![x\approx y]\\!]^{\bf 2}$, by
Lemma 5.1 (iv). So, $x\in C$ as desired.

$\square$

###### Corollary 5.5

Suppose that $u\in{\mathbf{V}}^{\bf 2}$ is such that ${\mathbf{V}}^{\bf
2}\vDash u\not=\emptyset$ and let $v$ be a core for $u$. Then for any
$x\in{\mathbf{V}}^{\bf 2}$ there is $y\in v$ such that $[\\![x\approx
y]\\!]^{\bf 2}=[\\![x\in u]\\!]^{\bf 2}$.

Proof.  It immediately follows from Lemma 5.2 and taking $\phi(x)$:= “$x\in
u$”. $\square$

In order to prove that the Axiom of Choice is true in ${\mathbf{V}}^{\bf 2}$,
it is enough to verify the set-theoretically equivalent principle Zorn’s
Lemma. Recall now that a partially ordered set $X$ is said to be inductive if
every chain in $X$ has upper bounds. Zorn’s Lemma states that any non-empty
inductive partially ordered set has a maximal element.

###### Lemma 5.6

Zorn’s Lemma is true in ${\mathbf{V}}^{\bf 2}$.

Proof.  Let $x,\leq_{x}\in{\mathbf{V}}^{\bf 2}$ where $x$ is a set and
$\leq_{x}$ an ordered relation on $x$. Let us now suppose the formula
$\phi(x)$ is given by ”$(x,\leq_{x})$ is a non-empty inductive partially
ordered set” and the formula $\psi(x)$ given by ”$(x,\leq_{x})$ has a maximal
element”.

From Lemma 5.2(ii), we only prove that: there is $u\in{\mathbf{V}}^{\bf 2}$
such that the condition ${\mathbf{V}}^{\bf 2}\vDash\phi(u)$ implies
${\mathbf{V}}^{\bf 2}\vDash\psi(u)$. Indeed, let us suppose $y$ is a core for
$x$ and take the relation $\leq_{y}$ on $y$ defined by “$z\leq_{y}z^{\prime}$
iff $[\\![z\leq_{X}z^{\prime}]\\!]=1$” for $z,z^{\prime}\in y$. It is not hard
to see that $\leq_{y}$ is partial ordering on $y$. We will prove that every
chain in $y$ has upper bound in it. Indeed, let $C$ a chain in $y$ and take
$C^{\prime}=\\{\langle c,1\rangle:c\in C\\}$. Then, it is clear that
${\mathbf{V}}^{\bf 2}\vDash C^{\prime}\,\,\text{\em is a chain in}\,\,x$. From
the latter and Theorem 3.6 applyied to $\bf 2$, we have that there is
$u\in{\mathbf{V}}^{\bf 2}$ such that ${\mathbf{V}}^{\bf 2}\vDash
u\,\,\text{\em is a upper bound for}\,\,C^{\prime}\,\,\text{\em in}\,\,x$.
Taking $w\in y$ such that $[\\![w\approx u]\\!]^{\bf 2}=1$. Thus, $w$ is an
upper bound for $C$ in $y$. For every $x\in C$, we have $[\\![x\in
C^{\prime}]\\!]^{\bf 2}=1$. Then, $[\\![x\leq_{x}u]\\!]^{\bf 2}=1$ and so
$[\\![x\leq_{x}w]\\!]^{\bf 2}=1$. Therefore, $x\leq_{y}w$ and then $y$ is
inductive. By Zorn’s Lemma in ${\mathbf{V}}^{\bf 2}$, $y$ has maximal element
$c$. We claim $[\\![c\in x]\\!]^{\bf 2}=1$ and ${\mathbf{V}}^{\bf 2}\vDash
c\,\,\text{\em is a maximal element of}\,\,x$. Indeed, let us take
$t\in{\mathbf{V}}^{\bf 2}$ and apply Corollary 5.5, there is $z\in y$ such
that $[\\![t\in x]\\!]^{\bf 2}=[\\![t\approx z]\\!]^{\bf 2}$. So,
$[\\![c\leq_{x}t\wedge t\in x]\\!]^{\bf 2}=[\\![c\leq_{x}t\wedge t\approx
z]\\!]^{\bf 2}\leq[\\![c\leq_{x}z]\\!]^{\bf 2}$, by Leibniz’s law. Let us now
$u$ the mixture $a\cdot z+a^{*}\cdot c$ where $a=[\\![c\leq_{x}t]\\!]^{\bf 2}$
and $a^{*}$ is the Boolean complement of $a$. Then, by Mixing Lemma we have
$a\leq[\\![t\approx u]\\!]^{\bf 2}$ and $a^{*}\leq[\\![c\approx u]\\!]^{\bf
2}$. So, $[\\![u\in x]\\!]^{\bf 2}=1$. From the latter and by the definition
of $y$, there is $k\in y$ such that $[\\![k\approx u]\\!]^{\bf 2}=1$. On the
other hand, it is not hard to see that $[\\![c\leq_{x}u]\\!]^{\bf 2}=1$. So,
by Leibniz law, we have $[\\![c\leq_{x}k]\\!]^{\bf 2}=1$ and then
$c\leq_{y}k$. Hence, $c=k$ by the maximality of $c$. Therefore,
$[\\![c\leq_{x}z]\\!]^{\bf 2}\leq[\\![c\approx z]\\!]^{\bf 2}$ and so
$[\\![c\leq_{x}t\wedge t\in x]\\!]^{\bf 2}\leq[\\![t\approx c]\\!]^{\bf 2}$.
Thus, ${\mathbf{V}}^{\bf 2}\vDash\forall t\in x[(c\leq_{x}t)\to(t\approx c)]$
as desired. $\square$

###### Theorem 5.7

All set-theoretic axioms $ZF_{C_{1}}$ plus axiom of choice (AC) are valid in
${\mathbf{V}}^{\bf 2}$.

Proof.  The proof of the validity of axioms (Pairing), (Colletion),
(Separation), (Empty set), (Union), (Infinity) and (Induction) are the same
the ones given in Section 3.1, using Leibniz law and taking the Boolean
algebra 2. The validity of axioms (Extensionality), (Powerset) and (Empty set)
immediately follows from Theorem 4.5. The validity of (AC) follows form Lemma
5.6 and the fact that on $ZF_{C_{1}}$ the Law of Excluded Middle holds.
$\square$

It is clear that for all $u\in{\mathbf{V}}^{\bf 2}$ and every formula
$\phi(x)$, we can always take $[\\![\neg\psi(u)]\\!]^{\bf 2}=1$ and
$[\\![\circ\psi(u)]\\!]^{\bf 2}=0$, it is possible to see that Leibniz law is
hold. These conditions work perfectly for induction over the complexity of the
formula; and more interestingly, ${\mathbf{V}}^{\bf 2}$ is paraconsistent. To
see this, it is enough to take the formula $\phi(x):=x\approx x$. So,
${\mathbf{V}}^{\bf 2}$ is a paraconsistent of ZFC.

##  6 $F$-structure valued models for $C_{1}$ Paraconsistent Set Theory

In this section, we will construct the class ${\mathbf{V}}^{\langle{\bf
A},N,O\rangle}$ of $C_{1}$-structure valued model over $\langle{\bf
A},N,O\rangle$ following the methodology developed for Heyting-valued models.
By ${\cal L}_{\in}$, we denote the first-order language of set theory which
consists of the propositional connectives $\\{\to,\wedge,\vee,\neg\\}$ of the
$C_{1}$ and two binary predicates $\in$ and $\approx$. As it was done in
Section 5, we will consider formulas with $\circ$. But clearly, taking the
language $\\{\to,\wedge,\vee,\neg\\}$ for our system, it works perfectly with
formulas with $\circ$ because $\circ\alpha=\neg(\alpha\wedge\neg\alpha)$. We
can expand this language by adding all the elements of
${\mathbf{V}}^{\langle{\bf A},N\rangle}$; the expanded language we will denote
${\cal L}_{\langle{\bf A},N,O\rangle}$. Now, we will define a valuation by
induction on the complexity of a closed formula in ${\cal L}_{\langle{\bf
A},N,O\rangle}$. Then, for a given complete and saturate N4-structure
$\langle{\bf A},N,O\rangle$, the mapping $[\\![\cdot]\\!]^{\langle{\bf
A},N,O\rangle}:{\cal L}_{\langle{\bf A},N,O\rangle}\to\langle{\bf
A},N,O\rangle$ is defined as follows:

$\displaystyle[\\![u\in v]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle=\bigvee\limits_{x\in dom(v)}(v(x)\wedge[\\![x\approx
u]\\!]^{\langle{\bf A},N,O\rangle});$ ($C_{1}$-1) $\displaystyle[\\![u\approx
v]\\!]^{\langle{\bf A},N\rangle}$ $\displaystyle=\bigwedge\limits_{x\in
dom(u)}(u(x))\to[\\![x\in v]\\!]^{\langle{\bf
A},N\rangle})\wedge\bigwedge\limits_{x\in dom(v)}(v(x)\to[\\![x\in
u]\\!]^{\langle{\bf A},N,O\rangle});$ ($C_{1}$-2)
$\displaystyle[\\![\varphi\\#\psi]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle=[\\![\varphi]\\!]^{\langle{\bf
A},N,O\rangle}\\#[\\![\psi]\\!]^{\langle{\bf A},N\rangle}\,\,\text{for
every}\,\,\\#\in\\{\wedge,\vee,\to\\};$ ($C_{1}$-3)
$\displaystyle[\\![\neg\varphi]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle\in N_{[\\![\varphi]\\!]^{\langle{\bf
A},N,O\rangle}}\,\,\text{and}\,\,[\\![\neg\neg\psi)]\\!]^{\langle{\bf
A},N,O\rangle}\leq[\\![\psi)]\\!]^{\langle{\bf A},N,O\rangle};$ ($C_{1}$-4)
$\displaystyle[\\![\circ\varphi]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle\in O_{[\\![\varphi]\\!]^{\langle{\bf A},N,O\rangle}};$
($C_{1}$-5) $\displaystyle[\\![\circ\alpha]\\!]^{\langle{\bf
A},N,O\rangle}\wedge[\\![\circ\beta]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle\leq[\\![\circ(\alpha\wedge\beta)]\\!]^{\langle{\bf
A},N,O\rangle}\wedge[\\![\circ(\alpha\vee\beta)]\\!]^{\langle{\bf
A},N,O\rangle}\wedge[\\![\circ(\alpha\rightarrow\beta)]\\!]^{\langle{\bf
A},N,O\rangle};$ ($C_{1}$-6) $\displaystyle[\\![\exists
x\varphi]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle=\bigvee\limits_{{u\in{\mathbf{V}}^{\langle
A,N,O\rangle}}}[\\![\varphi(u)]\\!]^{\langle{\bf
A},N\rangle}\,\,\text{and}\,\,[\\![\forall x\varphi]\\!]^{\langle{\bf
A},N,O\rangle}=\bigwedge\limits_{{u\in{\mathbf{V}}^{\langle
A,N\rangle}}}[\\![\varphi(u)]\\!]^{\langle{\bf A},N,O\rangle};$ ($C_{1}$-7)
$\displaystyle[\\![u\approx v]\\!]^{\langle{\bf A},N,O\rangle}$
$\displaystyle\leq[\\![\\#\phi(u)]\\!]^{\langle{\bf
A},N,O\rangle}\to[\\![\\#\phi(v)]\\!]^{\langle{\bf A},N,O\rangle}\,\,\text{for
any formula}\,\,\phi\,\,\text{and with}\,\,\\#\in\\{\neg,\circ\\}.$
($C_{1}$-8)

$[\\![\varphi]\\!]^{\bf 2}$ is called the truth-value of the sentence
$\varphi$ in the language ${\cal L}_{\langle{\bf A},N,O\rangle}$ in the f bi-
valuation valued mode over the Boolean algebra $\bf 2$. Now, we say that a
sentence $\varphi$ in the language ${\cal L}_{\langle{\bf A},N,O\rangle}$ is
said to be valid in ${\mathbf{V}}^{\langle{\bf A},N,O\rangle}$, which is
denoted by ${\mathbf{V}}^{\langle{\bf A},N,O\rangle}\vDash\varphi$, if
$[\\![\varphi]\\!]^{\langle{\bf A},N,O\rangle}=1$.

The following Lemma can be proved in a similar way to Lemma 5.1 of Section 5.

###### Lemma 6.1

The following properties true in ${\mathbf{V}}^{\langle{\bf A},N,O\rangle}$:

  * (i)

$[\\![u\approx u]\\!]^{\langle{\bf A},N,O\rangle}=1$,

  * (ii)

$u(x)\leq[\\![x\in u]\\!]^{\langle{\bf A},N,O\rangle}$ for every $x\in
dom(u)$,

  * (iii)

$[\\![u\approx v]\\!]^{\langle{\bf A},N,O\rangle}=[\\![v\approx
u]\\!]^{\langle{\bf A},N,O\rangle}$, for every
$u,v\in{\mathbf{V}}^{\langle{\bf A},N,O\rangle}$

  * (iv)

$[\\![u\approx v]\\!]^{\langle{\bf
A},N,O\rangle}\leq[\\![\phi(u)\to\phi(v)]\\!]^{\langle{\bf A},N,O\rangle}$ for
any formula $\phi$ (Leibniz’s law).

Taking in mind the proofs of Section 5, particularly, Lemmas 5.2 and 5.4,
Collary 5.5 and Lemma 5.6. Now, replacing 2 for an every $C_{1}$-structure
${\langle{\bf A},N,O\rangle}$, we have proved the following Theorem:

###### Theorem 6.2

All set-theoretic axioms $ZF_{C_{1}}$ plus axiom of choice (AC) are valid in
${\mathbf{V}}^{\langle{\bf A},N,O\rangle}$.

Al the proofs given in this section can be adapted easily to the context of da
Costa’s $C_{n}$ and the Logics of Formal Inconsistency, [6].

###  6.1 $F$-structure valued models for $ZF_{C_{\omega}}$

Under the same consideration to $C_{1}$, we can present the $F$-structure
valued models for $ZF_{C_{\omega}}$. But now the mapping
$[\\![\cdot]\\!]^{\langle{\bf A},N\rangle}:{\cal L}_{\langle{\bf
A},N\rangle}\to\langle{\bf A},N\rangle$ is defined from ($C_{1}$-1) to
($C_{1}$-4), ($C_{1}$-7) and ($C_{1}$-8). Then, the following Theorem can be
proved analogously to Corollary 4.6:

###### Theorem 6.3

(Figallo-Orellano & Slagter [15]). All set-theoretic axioms $ZF_{C_{\omega}}$
are valid in ${\mathbf{V}}^{\langle{\bf A},N\rangle}$.

In the book [3, pag. 70], it was presented a proof that Zorn’s Lemma is valid
on every Heyting-valued models. The form of the this Lemma, which is taken in
this book, is in fact is stronger version but still strong enough to prove
(AC). Adapting the proofs for intuitionistic case and using Leibniz’s law, we
have proved the following Lemma.

###### Lemma 6.4

Zorn’s Lemma is true in ${\mathbf{V}}^{\langle{\bf A},N\rangle}$ and so (AC)
is true in ${\mathbf{V}}^{\langle{\bf A},N\rangle}$.

##  7 Conclusions and future works

The problem of independence of Continuum Hypothesis (HC) from
$ZF_{C_{\omega}}$ can be treated using Boolean-valued models. It is clear that
Boolean-valued models validate the axioms of $C_{\omega}$; and, the
independence problem is one of the meta-theory of the system. So, the
classical proof could be used to see that (HC) is independent of
$ZF_{C_{\omega}}$. The paraconsistent view of these affairs will be part of
our future investigation; moreover, we are interested developing a technique
obtain a paraconsistent forcing.. In particular, we will interested on
paraconsistent set theories where Boolean-valued models are not models of
these theories.

On the other hand, in this note we have presented some paraconsistent set
theories where the models validate Choice Axiom (CA). In this setting, we
think the problem of the independence in different PSTs is very interesting.

## References

  * [1] A. Almukdad and D. Nelson. Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231-233, 1984.
  * [2] A. Balbes and P. Dwinger. Distributive lattices. Univ. of Missouri Press, Columbia, 1974.
  * [3] J. Bell. Zorn’s lemma and complete Boolean algebras in intuitionistic type theories. J. Symbolic Logic. 62, 1265–1279, 1997.
  * [4] J. Bell. Set theory. Boolean valued models and independence proofs. Oxford Science Pubblications, 2005.
  * [5] J. Bell. Intuitionistic set theory. College Publications. 2014.
  * [6] W. Carnielli, M. Coniglio, and J. Marcos. Logics of Formal Inconsistency. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 14, pages 1–93. Springer, 2nd edition, 2007.
  * [7] N. da Costa. On the theory of inconsistent formal systems (In Portuguese). Habilitation thesis, Curitiva. Editora UFPR, Brazil, 1963.
  * [8] N. da Costa. On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, vol. 15, 497–510, 1974.
  * [9] N da Costa and E. Alves. A semantical analysis of the calculi $C_{n}$. Notre Dame Journal of Formal Logic, 18, 621–30, 1977.
  * [10] F. Esteva, A. Figallo-Orellano, T. Flaminio and L. Godo. Logics of Formal Inconsistency Based on Distributive Involutive Residuated Lattices, J. Log. Comput., 31(5):1226–1265, 2021.
  * [11] F. Esteva, A. Figallo-Orellano, T. Flaminio and L. Godo, Some categorical equivalences for Nelson algebras with consistency operators. Proceeding of the 12th Conference of the European for Fuzzy Logic and Technology (EUSFLAT), Atlantis Studies in Uncertainty Modelling, vol. 3 (2021), pp. 420-426
  * [12] M. Fidel. The decidability of the calculi $C_{n}$. Reports on Mathematical Logic, 8:31–40, 1977.
  * [13] Aldo Figallo-Orellano, Miguel Pérez-Gaspar and Everardo Bárcenas, Fidel semantics for propositional and first-order version of the logic of CG’3. Logic and Logical Philosophy, 2022.
  * [14] A. Figallo-Orellano, M. Pérez-Gaspar and J. M. Ramirez-Contreras, Paraconsistent and Paracomplete logics based based on $k$-cyclic modal pseudocomplemented De Morgan algebras. Studia Logica, 2022.
  * [15] A. Figallo-Orellano and J. Slagter, Models for da Costa’s paraconsistent set theory, Workshop Brasileiro de Lógica (WBL), I . Sociedade Brasileira de Computação, 2020 . p. 33-40.
  * [16] A. Figallo-Orellano and J. Slagter, Monteiro’s algebraic notion of maximal consistent theory for Tarskian logics, Fuzzy Sets and Systems, 2022.
  * [17] M. Fitting. Intuitionistic Logic, Model Theory and Forcing. North–Holland Publ. Comp., Amsterdam, 1969. (Ph. D. Thesis)
  * [18] M. Osorio, A. Figallo-Orellano and M. Pérez-Gaspar. A family of genuine and non-algebraisable C-systems. Journal of Applied Non-Classical Logics, 31:1, 56–84, 2021
  * [19] S. Odintsov. Algebraic Semantics for Paraconsistent Nelson’s Logic. J. Logic Computat., Vol. 13 No. 4, 2003.
  * [20] H. Rasiowa, An algebraic approach to non-clasical logics, Studies in logic and the foundations of mathematics, vol. 78. North-Holland Publishing Company, Amsterdam and London, and American Elsevier Publishing Company, Inc., New York, 1974.
  * [21] R. Wójcicki. Lectures on propositional calculi. Ossolineum, Warsaw, 1984.

A. Figallo-Orellano
Departamento de Matemática
Universidad Nacional del Sur (UNS)
Bahía Blanca, Argentina
e-mails<EMAIL_ADDRESS>


# Energy and Adjacency Spectra of Semigraphs

Pralhad M. Shinde1, Charusheela Deshpande2 1,2Dept. of Maths, College of
Engineering Pune, Maharashtra-411005, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the energy of semigraphs and obtain some bounds, and
show that one of the bounds is tight. We also study the spectra of the
adjacency matrix of a special type of rooted 3-uniform semigraph and enumerate
those explicitly.

Keywords: Adjacency matrix of semigraph, Energy of semigraph

AMS classification: 05C99; 05C50

##  1\. Introduction

If $G$ is a graph of order $n$ and $A$ is its adjacency matrix then energy of
graph $G$, denoted by $\mathcal{E}(G)$, is the sum of absolute values of its
eigenvalues. Energy of graph is extensively studied in literature [2]. In [1],
authors studied the energy of semigraphs for their adjacency matrix which is
not a symmetric matrix. In [3], the first author defined the adjacency matrix
which enjoys the symmetric property and showed that various spectral graph
theory results can be extended to semigraphs. And it draws the inspiration to
study the energy of semigraph with the new definition of adjacency matrix.

This paper is organized as follows. In Section 2, we study eigenvalues of
special type of rooted 3-uniform semigraph. In Section 3, we define the energy
of a semigraph and give lower and upper bounds for the same. Further, we
enumerate energies of some special type of semigraphs and compare them with
the bounds obtained.

## Preliminaries

For all basic definitions and standard notations please refer to [3], [5].
We recall some definitions here;

######  Definition 1.1.

Let $V$ be a non-empty set having $n$ elements. A $semigraph$ is a pair
G=$(V,E),$ where the elements of $V$are called vertices and $E$ is a set of
ordered $k$-tuples of distinct vertices, whose elements are called edges of
$G$; for $n\geq 2,$ satisfying the following conditions:

  1. (1)

Any two edges have at most one vertex in common

  2. (2)

Two edges $(u_{1},u_{2},\cdots,u_{k})$ and $(v_{1},v_{2},\cdots,v_{r})$ are
considered to be equal if

    1. (a)

r = k and

    2. (b)

either $u_{i}=v_{i}$ for $1\leq i\leq k,$ or $u_{i}=v_{k-i+1}$, for $1\leq
i\leq k.$

Thus the edge $(u_{1},u_{2},\cdots,u_{r})$ is same as
$(u_{r},u_{r-1},\cdots,u_{1})$.

Two vertices $v_{i},\;v_{j}$ in a semigraph are said to be adjacent if they
belong to the same edge and are said to be consecutively adjacent if in
addition they are consecutive in order as well. For the edge
$e=(u_{1},u_{2},\cdots,u_{n})$, $u_{1}$ and $u_{n}$ are called the end
vertices of $e$ and $u_{2},u_{3},\cdots,u_{n-1}$ are called the middle
vertices of $e$. Note that $u_{i},\;u_{j}$ are adjacent for all $1\leq i,j\leq
n$ while $u_{i},u_{i+1}$ are consecutively adjacent for all $1\leq i\leq n-1$.

For a semigraph, we define following types of vertices and edges:

  1. (1)

$u_{i}$ is said to be a pure end vertex if it is an end vertex of every edge
to which it belongs.

  2. (2)

$u_{i}$ is said to be a pure middle vertex if it is a middle vertex of every
edge to which it belongs.

  3. (3)

$u_{i}$ is said to be a middle end vertex if it is middle vertex of at least
one edge and end vertex of at least one other edge.

  4. (4)

An edge $e=(u_{1},u_{2},\cdots,u_{k}),\;k\geq 2$ is said to be full edge if
$u_{1}$ and $u_{k}$ are pure end vertices.

  5. (5)

An edge $e=(u_{1},u_{2},\cdots,u_{k}),\;k>2$ is said to be an half edge if
either $u_{1}$ or $u_{k}$ (or both) are middle end vertices.

  6. (6)

An edge $e=(u_{1},u_{2})$ is said to be a quarter edge if both $u_{1}$ and
$u_{2}$ are middle end vertices while $e=(u_{1},u_{2})$ will be half edge if
exactly one of $u_{1}$ and $u_{2}$ is a middle end vertex and other is a pure
end vertex.

For a full edge $e=(u_{1},u_{2},\cdots,u_{k})$,
$(u_{i},u_{i+1})\;\forall\;1\leq i\leq k-1$ is called a partial edge of $e$
while for a half edge $e=(u_{1},u_{2},\cdots,u_{k-1},u_{k})$, $(u_{1},u_{2})$
is called partial half edge if $u_{1}$ is middle end vertex, $(u_{k-1},u_{k})$
is a partial half edge if $u_{k}$ is middle end vertex and
$(u_{i},u_{i+1})\;\forall\;2\leq i\leq k-2$ are partial edges. Thus, any half
edge can have at most two partial edges.

######  Example 1.2.

Let $G=(V,E)$ be a semigraph, with $V=\\{v_{1},v_{2},\cdots,v_{10}\\}$ as a
vertex set and $E=\\{(v_{1},v_{2},v_{3},v_{4},v_{5}),$ $(v_{1},v_{7},v_{8}),$
$(v_{2},v_{6},v_{8}),$ $(v_{1},v_{9}),$ $(v_{6},v_{7})\\}.$ as an edge set.

$v_{9}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{8}$$v_{7}$$v_{10}$ Figure
1.

In Fig. 1, vertices $v_{1},v_{5},v_{8},$and$v_{9}$ are the pure end vertices;
$v_{3},v_{4}$ are pure middle vertices; $v_{2}$,$v_{6},$and$v_{7}$ are the
middle end vertices and $v_{10}$ is an isolated vertex. Further,
$(v_{1},v_{9})$, $(v_{1},v_{7},v_{8})$, $(v_{1},v_{2},v_{3},v_{4},v_{5})$ are
full edges whereas $(v_{2},v_{6},v_{8})$ is an half edge with only
$(v_{2},v_{6})$ as a partial half edge. Note that $(v_{6},v_{7})$ is the a
quarter edge.

######  Definition 1.3.

A semigraph $G=(V,E)$ is said to be connected if for any two vertices
$u,\;v\;\in E$, there exist a sequence of edges $e_{i_{1}},\cdots,e_{i_{p}}$
for some $p$ such that $u\in e_{i_{1}},\;v\in e_{i_{p}}$ and $|e_{i_{j}}\cap
e_{i_{j+1}}|=1,\;\forall\;1\leq j\leq p-1$.

######  Notation 1.4.

Throughout this paper, we assume that semigraph is connected and $G=(V,E)$
denotes the semigraph with $n$ vertices and $m$ edges such that

  * •

$m_{1}$ is the number of full edges

  * •

$m_{2}$ is the number quarter edges

  * •

$m_{3}$ is the number of half edges with one partial half edge

  * •

$m_{4}$ is the number of half edges with two partial half edges

Note that $m=m_{1}+m_{2}+m_{3}+m_{4}$ and if $G$ is a graph then
$m_{2}=m_{3}=m_{4}=0$ and $m=m_{1}$.

##  2\. Adjacency matrix

Let $G$=$(V,E)$ be a semigraph, with $V=\\{v_{1},v_{2},\cdots,v_{n}\\}$ as a
vertex set and $E=\\{e_{1},e_{2},\cdots,e_{m}\\}$ as an edge set. Recall that
the graph skeleton [3, definition 1.5] of $G$ is an underlined graph structure
$G^{S}$ of the semigraph on $V$, where two vertices $v_{i}$, $v_{j}$ are
adjacent in $G^{S}$ iff $v_{i}$ and $v_{j}$ are consecutively adjacent in $G.$
Let $u_{i},u_{j}\in e=(u_{1},u_{2},\cdots,u_{k})$ for some $e\in E$. Let
$d_{e}(u_{i},u_{j})$ denote the distance between $u_{i}$ and $u_{j}$ in the
graph skeleton of $e$. The distance $d_{e}(u_{i},u_{j})$ is well-defined as
each pair of vertices in semigraph belongs to at most one edge.

######  Definition 2.1.

[3] We index the rows and columns of a matrix $A=(a_{ij})_{n\times n}$ by
vertices $v_{1},v_{2},\cdots,v_{n},$ where $a_{ij}$ is given as follows:

$a_{ij}=\begin{cases}d_{e}(v_{i},v_{j}),&\text{if $v_{i},\;v_{j}$ belong to a
full edge or a half edge such that }\\\ &\text{$(v_{i},v_{j})$ is neither a
partial half edge nor a quarter edge}\\\ \;\;\;\frac{1}{2},&\text{if
$(v_{i},\;v_{j})$ is a partial half edge}\\\ \;\;\;\frac{1}{4},&\text{if
$(v_{i},v_{j})$ is a quarter edge}\\\ \;\;\;0,&\text{otherwise}\end{cases}$

The matrix $A=(a_{ij})_{n\times n}$ is called the adjacency matrix of
semigraph $G.$

Let $A_{i}$ be the $i^{th}$ row of the adjacency matrix $A$, we define the
degree of vertices in semigraph as $d_{i}=A_{i}\mathbf{1}$, $\mathbf{1}$ being
a column matrix with all entries 1.

##  3\. Spectra of rooted 3-uniform semigraph tree

In this section, we compute the eigenvalues of the rooted 3-uniform semigraph
tree. Let $T^{3}_{n}$ denote the semigraph on $2n+1$ vertices with $n$ edges.
The edge set $E$ is given by $\\{(v_{1},v_{2i},v_{2i+1})\;\big{|}\;1\leq i\leq
n\\}$

$v_{1}$$v_{2n}$$v_{2n+1}$$v_{2}$$v_{3}$$v_{2r}$$v_{2r+1}$

$T^{3}_{n}$

Figure 2.

The adjacency matrix $A$ of rooted 3-uniform semigraph tree: $T^{3}_{n}$ is

$\bordermatrix{&{v_{1}}&{v_{2}}&{v_{3}}&{v_{4}}&{v_{5}}&\cdots&{v_{2n-2}}&{v_{2n-1}}&{v_{2n}}&{v_{2n+1}}\cr
v_{1}&0&1&2&1&2&\cdots&1&2&1&2\cr v_{2}&1&0&1&0&0&\cdots&0&0&0&0\cr
v_{3}&2&1&0&0&0&\cdots&0&0&0&0\cr v_{4}&1&0&0&0&1&\cdots&0&0&0&0\cr
v_{5}&2&0&0&1&0&\cdots&0&0&0&0\cr\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\cr
v_{2n-2}&1&0&0&0&0&\cdots&0&1&0&0\cr v_{2n-1}&2&0&0&0&0&\cdots&1&0&0&0\cr
v_{2n}&1&0&0&0&0&\cdots&0&0&0&1\cr v_{2n+1}&2&0&0&0&0&\cdots&0&0&1&0}\qquad$

######  Lemma 3.1.

The spectrum of $T^{3}_{n}$ is:

$\begin{pmatrix}-1&1&\lambda_{1}&\lambda_{2}&\lambda_{3}\\\
n-1&n-1&1&1&1\end{pmatrix}$

where $\lambda_{1},\lambda_{2},\lambda_{3}$ are roots of
$\lambda^{3}-(5n+1)\lambda-4n$.

###### Proof.

Consider the characteristic polynomial $T_{n}(\lambda)=det(\lambda I-A)$

$T_{n}(\lambda)=\begin{vmatrix}\lambda&-1&-2&-1&-2&\cdots&-1&-2&-1&-2\\\
-1&\lambda&-1&0&0&\cdots&0&0&0&0\\\ -2&-1&\lambda&0&0&\cdots&0&0&0&0\\\
-1&0&0&\lambda&-1&\cdots&0&0&0&0\\\ -2&0&0&-1&\lambda&\cdots&0&0&0&0\\\
&&&\vdots&&\vdots&&&&\\\ -1&0&0&0&0&\cdots&\lambda&-1&0&0\\\
-2&0&0&0&0&\cdots&-1&\lambda&0&0\\\ -1&0&0&0&0&\cdots&0&0&\lambda&-1\\\
-2&0&0&0&0&\cdots&0&0&-1&\lambda\end{vmatrix}$

Using co-factor expansion along the last column, we get

$\displaystyle T_{n}(\lambda)=$ $\displaystyle\lambda
A_{(2n+1)(2n+1)}+1A_{(2n)(2n+1)}-2A_{1(2n+1)}$

Further using the cofactor expansions of the above terms along the last
columns and later taking the cofactor expansion along the last row if the only
non-zero entry is in the first column-last row, we get

$\displaystyle T_{n}=$
$\displaystyle\lambda^{2}T_{n-1}-\lambda(\lambda^{2}-1)^{n-1}-T_{n-1}-2(\lambda^{2}-1)^{n-1}-2(\lambda^{2}-1)^{n-1}-4\lambda(\lambda^{2}-1)^{n-1}$
$\displaystyle T_{n}=$
$\displaystyle(\lambda^{2}-1)T_{n-1}-5\lambda(\lambda^{2}-1)^{n-1}-4(\lambda^{2}-1)^{n-1}$

Applying the same formula for $T_{n-1}$ we get

$T_{n-1}=(\lambda^{2}-1)T_{n-2}-5\lambda(\lambda^{2}-1)^{n-2}-4(\lambda^{2}-1)^{n-2}$

Plugging this to the formula of $T_{n}$, we get

$T_{n}=(\lambda^{2}-1)\left[(\lambda^{2}-1)T_{n-1}-5\lambda(\lambda^{2}-1)^{n-1}-4(\lambda^{2}-1)^{n-1}\right]-5\lambda(\lambda^{2}-1)^{n-1}-4(\lambda^{2}-1)^{n-1}$

Simplifying it gives us

$T_{n}=(\lambda^{2}-1)^{2}T_{n-2}-5\times
2\;\lambda(\lambda^{2}-1)^{n-1}-4\times 2\;(\lambda^{2}-1)^{n-1}$

Continuing recursive substitution we get

$T_{n}=(\lambda^{2}-1)^{n-1}T_{1}-5(n-1)\;\lambda(\lambda^{2}-1)^{n-1}-4(n-1)\;(\lambda^{2}-1)^{n-1}$

Here, $T_{1}$ is a single edge of length 3 and it’s characteristics equation
is $\lambda^{3}-6\lambda-4$. Therefore,

$T_{n}=(\lambda^{2}-1)^{n-1}(\lambda^{3}-6\lambda-4)-5(n-1)\;\lambda(\lambda^{2}-1)^{n-1}-4(n-1)\;(\lambda^{2}-1)^{n-1}$

Further simplifying and re-arranging the terms we get

$T_{n}(\lambda)=(\lambda^{2}-1)^{n-1}\;(\lambda^{3}-(5n+1)\lambda-4n)$

Hence, the spectrum of $T^{3}_{n}$ is: -1 and 1 repeated $n-1$ times and roots
$\lambda_{1},\lambda_{2},\lambda_{3}$ of $\lambda^{3}-(5n+1)\lambda-4n$. ∎

##  4\. Bounds on energy

Energy of semigraph is defined as the sum of absolute values of its
eigenvalues. In [1], authors have calculated energy of semigraphs using their
adjacency matrix. We use our definition of adjacency matrix to find the energy
and also find some bounds. The bounds obtained here are similar to the energy
bounds of graphs and it turns out as generalization of graph bounds.
To prove the theorem that follow we need the following definition of cartesian
product of semigarphs and result on eigenvalues of the product.

######  Definition 4.1.

[5] Let $G_{1}=(V_{1},E_{1})$, $G_{2}=(V_{2},E_{2})$ be two semigraphs with
$|V_{1}|=m,\;|V_{2}|=n$. The cartesian product $G=G_{1}\times G_{2}$ of
$G_{1}$ and $G_{2}$ is a semigraph on the vertex set
$V=\\{(u_{i},v_{j})\;|\;u_{i}\in V_{1},\;v_{j}\in V_{2}\\}$. The edges are of
the form
$\left((u_{i},v_{i_{1}}),(u_{i},v_{i_{r}}),\cdots,(u_{i},v_{i_{r}})\right)$
for some edge $(v_{i_{1}},v_{i_{2}},\cdots,v_{i_{r}})$ of $G_{2}$ or
$\left((u_{i_{1}},v_{j}),(u_{i_{2}},v_{j}),\cdots,(u_{i_{p}},v_{j})\right)$
for some edge $(u_{i_{1}},u_{i_{2}},\cdots,u_{i_{p}})$ of $G_{1}$.

Note that adjacency matrix of $G_{1}\times G_{2}$ is of form $I_{m}\otimes
B+A\otimes I_{n}$, where $A_{m\times m}$ and $B_{n\times n}$ are adjacency
matrices of $G_{1}$ and $G_{2}$ respectively. Thus, eigenvalues of
$G=G_{1}\times G_{2}$ are of the form $\lambda_{i}+\mu_{j}$ where
$\lambda_{i},\;\mu_{j}$ are eigenvalues of $G_{1}$ and $G_{2}$ respectively.

######  Theorem 4.2.

If $\mathcal{E}(G)$ is a rational number then it must be either an even
integer or a rational number of the form $\frac{N}{2^{r}}$, for some
$N\in\mathbb{Z},\;r\in\mathbb{N}$ depending on whether $G$ has no middle end
vertices or has middle end vertices.

###### Proof.

Let $\lambda_{1},\lambda_{2},\cdots,\lambda_{n}$ be eigenvalues of adjacency
matrix $A$ of $G$. Trace of $A$ being $0$ implies
$\displaystyle\sum_{i=1}^{n}\lambda_{i}=0$.
Hence,$\displaystyle\sum_{i=1}^{k}\lambda_{i}=\displaystyle\sum_{i=k+1}^{n}\lambda_{i}$,
where $\lambda_{1},\lambda_{2},\cdots,\lambda_{k}$ are non-negative
eigenvalues and $\lambda_{k+1},\lambda_{k+2},\cdots,\lambda_{n}$ are negative
eigenvalues.
Thus, $\mathcal{E}(G)=2(\lambda_{1}+\lambda_{2}+\cdots+\lambda_{k})$. Note
that $\lambda_{1}+\lambda_{2}+\cdots+\lambda_{k}$ is an eigenvalue of $G\times
G\times\cdots\times G$(k-times). We consider two cases based on the presence
or absence of middle end vertices.
Case 1: If $G$ has no middle end vertices then characteristic polynomial of
adjacency matrix is a monic polynomial with integer coefficient, and rational
root of such polynomial must be an integer. Hence,
$\lambda_{1}+\lambda_{2}+\cdots+\lambda_{k}$ is an integer and
$\mathcal{E}(G)$ is an even integer.
Case 2: If $G$ has middle end vertices then adjacency matrix of $G$ contains
some entries as $\frac{1}{2},\,\frac{1}{4}$, so does the adjacency matrix of
$G\times G\times\cdots\times G$. Hence, the characteristic polynomial which is
monic has coefficients of the form $\frac{p}{2^{r}}$ for some
$p\in\mathbb{Z},\;r\in\mathbb{N}$. Let ${2^{l}}$ be the least common multiple
of all denominators, multiply the characteristics equation by it to make
coefficients integer. Hence, the rational roots are of the form
$\frac{N}{2^{r}}$, for some $N\in\mathbb{Z},\;r\in\mathbb{N}$. So, if
$\lambda_{1}+\lambda_{2}+\cdots+\lambda_{k}$ is a rational number then
$\mathcal{E}(G)$ must of the form $\frac{N}{2^{r}}$, for some
$N\in\mathbb{Z},\;r\in\mathbb{N}$.

∎

######  Remark 4.3.

When the semigraph $G$ is a graph then we get the graph theory result [4,
Theorem 3.27].

######  Theorem 4.4.

Let $|V|=n$, $|E|=m$, $r_{i}$ is the size of the $i^{th}$ edge, then

$\displaystyle\mathcal{E}(G)\leq\sqrt{n\left(\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}\right)}$

where $m_{2},m_{3},m_{4}$ are defined as earlier.

###### Proof.

Consider

$\mathcal{E}(G)^{2}=\left(\sum_{i=1}^{n}|\lambda_{i}|\right)^{2}$

by Cauchy-Schwarz inequality

$\mathcal{E}(G)^{2}=\left(\sum_{i=1}^{n}|\lambda_{i}|\right)^{2}\leq
n\sum_{i=1}^{n}|\lambda_{i}|^{2}$

by [3, lemma 3.3]

$\sum_{i=1}^{n}|\lambda_{i}|^{2}=\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}$

Hence, combining these two together we get

$\mathcal{E}(G)^{2}\leq
n\left(\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}\right)$

Hence,

$\displaystyle\mathcal{E}(G)\leq\sqrt{n\left(\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}\right)}.$

∎

######  Remark 4.5.

When the semigraph $G$ is a graph, then $m_{2}=m_{3}=m_{4}=0$ and
$r_{i}=2,\;\forall i$. Thus, we get [2, Theorem 5.1]

$\displaystyle\mathcal{E}(G)\leq\sqrt{n\left(\frac{1}{6}\sum_{i=1}^{m}2^{2}(2^{2}-1\right)}=\sqrt{n\left(\frac{1}{6}12m\right)}=\sqrt{2mn}$

######  Theorem 4.6.

Let $|V|=n$, $|E|=m$, $r_{i}$ is the size of the $i^{th}$ edge, then

$\displaystyle\mathcal{E}(G)\geq\sqrt{\frac{1}{3}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{4}m_{2}-\frac{3}{2}m_{3}-m_{4}}$

where $m_{2},m_{3},m_{4}$ are defined as earlier.

###### Proof.

We know that $trace(A)=0$, implies $\displaystyle\sum_{i=1}^{n}\lambda_{i}=0.$
Hence $\displaystyle\left(\sum_{i=1}^{n}\lambda_{i}\right)^{2}=0.$ Thus,

$\displaystyle\sum_{i=1}^{n}\lambda^{2}_{i}+2\sum_{i<j}\lambda_{i}\lambda_{j}=0$

by [3, lemma 3.3]

$\displaystyle
2\sum_{i<j}\lambda_{i}\lambda_{j}=-\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)+\frac{15}{8}m_{2}+\frac{3}{4}m_{3}+\frac{1}{2}m_{4}\;\;\;\;\;\;\cdots(1)$

As

$\displaystyle\mathcal{E}(G)^{2}=\sum_{i=1}^{n}|\lambda_{i}|^{2}+2\sum_{i<j}|\lambda_{i}\lambda_{j}|$

and

$\displaystyle
2\sum_{i<j}|\lambda_{i}\lambda_{j}|\geq\big{|}2\sum_{i<j}\lambda_{i}\lambda_{j}\big{|}$

implies

$\displaystyle
2\sum_{i<j}|\lambda_{i}\lambda_{j}|\geq\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}$

Thus,

$\displaystyle\mathcal{E}(G)^{2}\geq\frac{2}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{4}m_{2}-\frac{3}{2}m_{3}-m_{4}$

Hence,

$\displaystyle\mathcal{E}(G)\geq\sqrt{\frac{1}{3}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{4}m_{2}-\frac{3}{2}m_{3}-m_{4}}.$

∎

######  Remark 4.7.

When semigraph $G$ is a graph [2, Theorem 5.2], we get

$\displaystyle\mathcal{E}(G)\geq 2\sqrt{m}.$

######  Theorem 4.8.

Let $|V|=n$, $|E|=m$, $r_{i}$ is the size of the $i^{th}$ edge, then

$\displaystyle\mathcal{E}(G)\leq\lambda_{1}+\sqrt{(n-1)\left(\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}-\lambda^{2}_{1}\right)}$

where $m_{2},m_{3},m_{4}$ are defined as earlier and $\lambda_{1}$ is the
largest eigenvalue.

###### Proof.

By definition,

$\displaystyle\mathcal{E}(G)=\sum_{i=1}^{n}|\lambda_{i}|$

by Cauchy Schwarz inequality

$\displaystyle\sum_{i=2}^{n}|\lambda_{i}|\leq\sqrt{(n-1)\sum_{i=2}^{n}|\lambda_{i}|^{2}}\;\;\;\;\
\cdots(2)$

by [3, lemma 3.3]

$\sum_{i=1}^{n}|\lambda_{i}|^{2}=\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}$

implies

$\sum_{i=2}^{n}|\lambda_{i}|^{2}=\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}-\lambda^{2}_{1}$

Thus, by inequality (2)

$\displaystyle\sum_{i=2}^{n}|\lambda_{i}|\leq\sqrt{(n-1)\left(\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}-\lambda^{2}_{1}\right)}$

Hence,

$\displaystyle\mathcal{E}(G)\leq\lambda_{1}+\sqrt{(n-1)\left(\frac{1}{6}\sum_{i=1}^{m}r^{2}_{i}(r^{2}_{i}-1)-\frac{15}{8}m_{2}-\frac{3}{4}m_{3}-\frac{1}{2}m_{4}-\lambda^{2}_{1}\right)}$

∎

######  Remark 4.9.

When the semigraph $G$ is a graph then we get [2, Theorem 5.3],

$\displaystyle\mathcal{E}(G)\leq\lambda_{1}+\sqrt{(n-1)\left(2m-\lambda^{2}_{1}\right)}$

###  4.1. Energies of some special semigraphs

We list down the energies of families of a few semigraphs. Let $S^{3}_{2,n}$
denote a star semigraph having one edge of 3 vertices and n edges of 2
vertices and $S^{3}_{n}$ represent 3-uniform star semigraph on $2n+1$ vertices
and $n$ edges.

$v_{2}$$v_{1}$$v_{3}$$v_{4}$$v_{r}$$v_{n+3}$$v_{r+1}$$v_{1}$$v_{0}$$v_{2}$$v_{3}$$v_{2n-1}$$v_{2n}$$v_{4}$

$S^{3}_{2,n}\hskip 156.49014ptS^{3}_{n}$

Figure 3.

We recall the eigenvalues of $S^{3}_{2,n}$ and $S^{3}_{n}$.

######  Lemma 4.10.

[3, Lemma 4.1] The spectra of star semigraph $S^{3}_{2,3}$ are:

$\begin{pmatrix}0&-2&\lambda_{1}&\lambda_{2}&\lambda_{3}\\\
n-1&1&1&1&1\end{pmatrix}$

where $\lambda_{1},\lambda_{2},\lambda_{3}$ are roots of the cubic polynomial
$\lambda^{3}-2\lambda^{2}-\frac{n+8}{4}\;+\frac{n}{2}.$

######  Lemma 4.11.

[3, Lemma 4.2] The spectra of star semigraph $S^{3}_{n}$ are:

$\begin{pmatrix}-2&2&1-\sqrt{2n+1}&1+\sqrt{2n+1}\\\ n&n-1&1&1\end{pmatrix}$

Here, we enumerate the eigenvalues of $S^{3}_{2,n},\;S^{3}_{n},T^{3}_{n}\;$
for small values of $n$. All values in the table are rounded to two decimal
places.

n | 1 | 2 | 3 | 4
---|---|---|---|---
$\mathcal{E}(S^{3}_{2,n})$ | 5.89 | 6.20 | 6.46 | 6.69
$\mathcal{E}(S^{3}_{n})$ | 5.46 | 10.47 | 15.29 | 20
$\mathcal{E}(T^{3}_{n})$ | 5.46 | 9.27 | 12.66 | 15.85
Energy table

The bound in theorem 4.8 is tight as it is attained by $S^{3}_{n}$ when $n=4$.
In $S^{3}_{n}$, we have $m_{2}=0=m_{3}=m_{4}$ and when $n=4$ we have $m=4$
edges and 9 vertices. Also, $r_{i}=4,\;\forall 1\leq i\leq 4$ and largest
eigenvalue $\lambda_{1}$ is 4. Hence, putting these values in 4.8 we get

$\displaystyle\displaystyle\mathcal{E}(S^{3}_{n})\leq$
$\displaystyle\;4+\sqrt{(9-1)\left(\frac{1}{6}\sum_{i=1}^{4}3^{2}(3^{2}-1)-4^{2}\right)}\;\;=\;20$

Thus, energy table confirms that $S^{3}_{n}$ attains the bound in theorem 4.8.

## Conclusion

There is ample scope for study in this topic further. One could study
different types of energies associated with the semigarph on parallel lines
with graph and see that the results of graph theory are special cases of the
results obtained. Also, section 4.1 opens up an interesting question to study
the family of semigraphs which attains the bounds in theorems 4.4, 4.6, 4.8.

## References

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  * [2] Li Xueliang, Shi Yongtang and Gutman Ivan, Graph energy, Springer Science & Business Media, 2012.
  * [3] Pralhad M. Shinde, Adjacency spectra of semigraphs(accepted for publication), Discrete Mathematics, Algorithms and Applications, 2023. https://doi.org/10.1142/s1793830923500118.
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