1 Introduction and motivation

AOs Calvo et al. (2002), a triumphantly mathematic model for handling information fusion problems, have performed vital part in the process of solving various pragmatic issues (Godo and Sierra 1988; Mayor and Torrens 1993; Ureña et al. 2019; Masoudi et al. 2018; Nolasco et al. 2019; Lucca et al. 2017; Campomanes-Alvarez et al. 2018; Jurio et al. 2013; Qiao 2022c; Gómez et al. 2016; Qiao 2022d; Zhang et al. 2018). Meanwhile, driven by successful application in practical problems, AOs have also shown a flourishing situation in theoretical research, especially for the study involving their properties and constructions. For example, in 2014, Lopez-Molina et al. (2014) studied the bimigrativity and total bimigrativity of AOs. In 2018, Arias-García et al. (2018) studied extremal Lipschitz continuous AOs. Puerta and Urrutia (2018) gave characterisations of self-dual AOs. Kolesárová et al. (2018) showed the characterizations of k-additive AOs. Durante and Ghiselli Ricci (2018) investigated supermigrativity of AOs. In 2019, Zahedi Zahedi Khameneh and Kilicman (2019) proposed a new composition-based construction method for AOs. At the same time, Jin et al. (2019) gave generalized phi-transformations of AOs. In 2021, López et al. (2021) presented a new way to construct AOs by means of affine operators. In 2022, De Baets and De Meyer (2022) studied maximal directions for monotonicity of AOs.

Particularly, in many cases, the information we want to aggregate may not be able to be successfully transformed into the unit closed interval, so the research on AOs in lattice-valued case becomes more and more important and urgent. Up to now, there have been many discussions about AOs in the lattice-valued case. For instance, in 2011, Komorníková and Mesiar (2011) introduced and discussed certain types of AOs on bounded posets. In 2017, Karaçal and Mesiar (2017) provided an investigation for general lattice-valued AOs. In 2018, Halaš et al. (2018) showed generating set for the clone of idempotent lattice-valued AOs. In 2019, Halaš et al. (2019a, 2019b) discussed the generation for AOs in lattice-valued case. In 2020, Chajda et al. (2020) studied decomposability of AOs on direct products of posets. In the same year, Kurač et al. (2020) discussed transfer-stable AOs on finite lattices. In 2022, Mesiar et al. (2022) proposed a new method to construct AOs in lattice-valued case.

Beyond all that, the construction for AOs, especially for usual AOs, from the known ones, as a meaningful and interesting construction method for AOs, has been considered in the literature, such as the method of extending t-(co)norms and fuzzy negations in lattice-valued case proposed by Palmeira and Bedregal (2012) and Palmeira et al. (2014), the method of composing t-norms on the lattice of lattice-valued mappings shown by Lobillo et al. (2021), the method of extending quasi-overlap functions on bounded posets proposed by Qiao (2022a, b), and so on.

In this paper, inspired by the construction methodology shown above, we focus on the constructions of nAAOs on function spaces composed of all fuzzy sets with bounded posets as truth values set, which is independent with the known construction approaches of AOs on bounded posets because most all of the existing construction methods for AOs mainly focus on the direct constructions with the help of different lattice structures rather than the constructions via a family of known AOs defined on a bounded poset, for example, the decomposable construction methods of AOs on product posets/lattices proposed in Komorníková and Mesiar (2011), and Chajda et al. (2020), the composite construction methods of AOs on bounded lattices given in Karaçal and Mesiar (2017), the construction methods of AOs on bounded lattices shown in Kurač et al. (2020) on the basis of lattice structures and the specific construction methods of AOs like Möbius product-based constructions presented in Mesiar et al. (2022). In addition, it also provides a unified way of constructing the commonly used nAAOs on function spaces via a family of known ones and covers the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when underlying bounded poset is taken as the corresponding truth values set, respectively. On the other hand, compared with classical fuzzy sets, interval-valued fuzzy sets and type-2 fuzzy sets can describe more uncertainty and better deal with practical application issues involving in decision-making, image processing, artificial neural network, clustering, fuzzy inference, vehicle guidance system, edge detection, and so on. And thus, the theory achievements obtained herein possess tremendous positive new feasible applications of nAAOs in real problems, especially in expert systems, image processing, decision-making and so on. In other words, the main contributions of this paper are as follows: (1) from the theoretical point of view, the construction methods obtained not only effectively fills the blank of theoretical research on constructing nAAOs on the corresponding function spaces with the help of known nAAOs defined on a bounded poset, but also covers the corresponding construction methods of commonly used nAAOs; (2) from the applications perspective, the theoretical results obtained provide the possibility for mining new applications of nAAOs in real application problems.

Major general targets of this paper are as follows.

  1. 1.

    To give the construction method of obtaining nAAOs on the function space composed of the fuzzy sets taking any fixed bounded poset as the truth values set via a family of nAAOs on that bounded poset.

  2. 2.

    To study the equivalent characterization of that the nAAOs on the function spaces described above point 1, can be generated by the expansion of a family of nAAOs on the underlying bounded posets of those function spaces.

  3. 3.

    To investigate basic properties of nAAOs obtained via constructions on the function spaces described above point 1, which include idempotency, symmetry, associativity and bisymmetry.

Follow-up arrangements for this article are: Sect. 2 lists the needed concepts related to posets and AOs; Sect. 3 shows the construction method of nAAOs on function spaces; Sect. 4 investigates properties of such obtained nAAOs on function spaces; Last section concludes the research results.

2 Preliminaries

A bounded poset \((P,\le _P)\) is a poset with a greatest element and a smallest element (Gierz et al. 2003; Qiao 2022a). From now on, symbols P, S always represent bounded posets \((P, \le _P, 0_P, 1_P)\) and \((S, \le _S, 0_S, 1_S)\), respectively (Qiao 2022a, 2023).

For a nonempty set X, the mapping from X to P is called a P-fuzzy subset of X. \(P^X\) denotes family of all P-fuzzy subsets of X (Qiao 2023).

For any \(x\in X\) and \(a\in P\), \(a_x\) denotes the P-fuzzy subset of X defined, for all \(y\in X\), by

$$\begin{aligned} a_x(y)=\left\{ \begin{array}{ll} a,&{} \text{whenever}~~ y=x, \\ 0_P,&{} \text{whenever}~~ y\ne x, \end{array} \right. \end{aligned}$$

and \(a_X\) is P-fuzzy subset of X provided, for all \(y\in X\), by \(a_X(y)=a\). In particular, \(\bot _{0_P}\) and \(\top _{1_P}\) denote the P-fuzzy subsets of X given, for all \(y\in X\), by \(\bot _{0_P}(y)=0_P\) and \(\top _{1_P}(y)=1_P\), respectively (Qiao 2023).

For any \(\alpha , \beta \in P^X\), \(\alpha \preceq \beta\) if, for all \(x\in X\), \(\alpha (x)\le _P\beta (x)\). Then \(\left( P^X,\preceq \right)\) forms a bounded poset (also a function space) with the smallest element as \(\bot _{0_P}\) and the greatest element as \(\top _{1_P}\) (Qiao 2023).

Meanwhile, for \(P^n\) and \(\left( P^X\right) ^n\) \((n\in \mathbb{N}-\{0\})\), we appoint the following symbols and notations.

  • \(\overrightarrow{a}\) in \(P^n\) means that \(\overrightarrow{a}=(a_1,a_2,\ldots ,a_n)\) and \(a_i\in P~(i=1,2,\ldots ,n)\).

  • \(\overrightarrow{\zeta }\) in \(\left( P^X\right) ^n\) means that \(\overrightarrow{\zeta }=\left( \zeta _1,\zeta _2,\ldots ,\zeta _n\right)\) and \(\zeta _i\in P^X~(i=1,2,\ldots ,n)\).

  • \(\overrightarrow{a}^I\) in \(P^n\) means that \(\overrightarrow{a}^I=(a,a,\ldots ,a)\) and \(a\in P\).

  • \(\overrightarrow{a}_X\) in \(\left( P^X\right) ^n\) means that \(\overrightarrow{a}_X=((a_1)_X,(a_2)_X,\ldots ,(a_n)_X)\) and \(a_i\in P~(i=1,2,\ldots ,n)\).

  • \(\overrightarrow{\zeta }^I\) in \(\left( P^X\right) ^n\) means that \(\overrightarrow{\zeta }^I=\left( \zeta ,\zeta ,\ldots ,\zeta \right)\) and \(\zeta \in P^X\).

  • For any \(\overrightarrow{a},\overrightarrow{b}\in P^n\), \(\overrightarrow{a}\lessdot \overrightarrow{b}\) if \(a_i\le _P b_i~(i=1,2,\ldots ,n)\).

  • For any \(\overrightarrow{\zeta },\overrightarrow{\eta }\in \left( P^X\right) ^n\), \(\overrightarrow{\zeta }\precapprox \overrightarrow{\eta }\) means that \(\zeta _i\preceq \eta _i~(i=1,2,\ldots ,n)\).

Definition 2.1

(See Demirci (2006) and Komorníková and Mesiar (2011)) An operator \({{\mathscr{A}}}:P^n\longrightarrow P\) is referred as an nAAO on P if, for any \(\overrightarrow{a},\overrightarrow{b}\in P^n\), it holds that:

(\({{\mathscr{A}}}\)1):

\({{\mathscr{A}}} \left( \overrightarrow{0_P}^{I}\right) =0_P\);

(\({{\mathscr{A}}}\)2):

\({{\mathscr{A}}} \left( \overrightarrow{1_P}^{I}\right) =1_P\);

(\({{\mathscr{A}}}\)3):

\({{\mathscr{A}}} \left( \overrightarrow{a}\right) \le _P {{\mathscr{A}}}\left( \overrightarrow{b}\right)\) whenever \(\overrightarrow{a}\lessdot \overrightarrow{b}\).

Definition 2.1 degrades into the definitions of nAAOs on [0, 1] and interval-valued nAAOs when the bounded poset P degrades into [0, 1] and the set composed by all closed subintervals of [0, 1] endowed with the component-wise order, respectively (Demirci 2006). In addition, since \(\left( P^X,\preceq ,\bot _{0_P},\top _{1_P}\right)\) also is a bounded poset, it follows that an operator \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\) is an nAAO on \(P^X\) means that, \(\forall ~\overrightarrow{\zeta },\overrightarrow{\eta }\in \left( P^X\right) ^n\), \({\text{A}}\left( {\overrightarrow {{ \bot _{{0_{P} }} }} ^{I} } \right) = \bot _{{0_{P} }}\), \({\text{A}}\left( {\overrightarrow {{{ \top }_{{1_{P} }} }} ^{I} } \right) ={ \top }_{{1_{P} }}\) and \({\text{A}}\left( {\vec{\zeta }} \right) \preceq {\text{A}}\left( {\vec{\eta }} \right)\) whenever \(\overrightarrow{\zeta }\precapprox \overrightarrow{\eta }\).

The following example collects some nAAO on bounded posets, most of which come from Demirci (2006), Komorníková and Mesiar (2011), Karaçal and Mesiar (2017), Qiao (2022b).

Example 2.1

  1. (i)

    The operators \({{\mathscr{A}}}^*,{{\mathscr{A}}}_*:P^n\longrightarrow P\) given, for any \(\overrightarrow{a}\in P^n\), by

    $$\begin{aligned} {{\mathscr{A}}}^*\left( \overrightarrow{a}\right) =\left\{ \begin{array}{ll} 0_P&{} \text{if}~~ \overrightarrow{a}=\overrightarrow{0_P}^{I}, \\ 1_P&{} \text{otherwise} \end{array} \right. \end{aligned}$$

    and

    $$\begin{aligned} {{\mathscr{A}}}_*\left( \overrightarrow{a}\right) =\left\{ \begin{array}{ll} 1_P&{} \text{if}~~ \overrightarrow{a}=\overrightarrow{1_P}^{I}, \\ 0_P&{} \text{otherwise} \end{array} \right. \end{aligned}$$

    are two nAAOs on P.

  2. (ii)

    The operator \({{\mathscr{A}}}_{\min }:L^n\longrightarrow L\) provided, \(\forall ~\overrightarrow{a}\in L^n\), as

    $$\begin{aligned} {{\mathscr{A}}}_{\min }\left( \overrightarrow{a}\right) =\inf \{a_i:i=1,2,\ldots ,n\} \end{aligned}$$

    is an nAAO on any complete latticeFootnote 1L.

  3. (iii)

    For any bounded poset P with at least three elements \(0_P, 1_P\) and \(\maltese\), the operator \({{\mathscr{A}}}:P^n\longrightarrow P\) given, for any \(\overrightarrow{a}\in P^n\), by

    $$\begin{aligned} {{\mathscr{A}}}\left( \overrightarrow{a}\right) =\left\{ \begin{array}{ll} 0_P&{} \text{if}~~ \overrightarrow{a}=\overrightarrow{0_P}^{I}, \\ 1_P&{} \text{if}~~ \overrightarrow{a}=\overrightarrow{1_P}^{I}, \\ \maltese &{} \text{otherwise} \end{array} \right. \end{aligned}$$

    is an nAAO on P.

  4. (iv)

    Each commonly used nAAO (see, e.g., n-ary t-norm, uninorm, quasi-overlap function etc.) on arbitrary bounded poset P is an nAAO on that P.

Definition 2.2

(See Demirci (2006) and Komorníková and Mesiar (2011).) An nAAO \({{\mathscr{A}}}:P^n\longrightarrow P\) satisfies

  • idempotency if \({{\mathscr{A}}}\left( \overrightarrow{a}^I\right) =a\) for any \(a\in P\);

  • symmetry if \({{\mathscr{A}}}\left( \overrightarrow{a}\right) ={{\mathscr{A}}}\left( a_{p(1)}, a_{p(2)},\ldots , a_{p(n)}\right)\) for any \(\overrightarrow{a}\in P^n\) and all permutations \((p(1),p(2),\ldots ,p(n))\) of \((1,2,\ldots ,n)\).

An element \(a^0\in P\) is called an annihilator (Komorníková and Mesiar 2011) of nAAO \({{\mathscr{A}}}:P^n\longrightarrow P\) on P if \({{\mathscr{A}}}\left( \overrightarrow{a}\right) =a^0\) for any \(\overrightarrow{a}\in P^n\) with \(a_i=a^0\) for some \(i\in \{1,2,\ldots ,n\}\).

Notice that, the definitions of idempotency, symmetry and annihilator of nAAO \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\) on \(P^X\) can be derived corresponding to Definition 2.2 and the definition of annihilator of nAAO \({{\mathscr{A}}}:P^n\longrightarrow P\) on P, respectively, because of that \(\left( P^X,\preceq ,\bot _{0_P},\top _{1_P}\right)\) also is a bounded poset and the details are omitted here.

3 Construction of nAAOs on function spaces

Herein, we give construction method of nAAOs on the function space \(P^X\) composed of all P-fuzzy subsets of X taking arbitrary fixed bounded poset P as truth values set via a family of nAAOs on bounded poset P. After that, we present the notion of representable nAAOs on \(P^X\) and show the equivalent characterization of them.

Proposition 3.1

Let \(\left\{ {{\mathscr{A}}}_x:P^n\longrightarrow P\mid x\in X\right\}\) be a family of nAAOs on P. Then operator \(\overline{{{\mathscr{A}}}}:\left( P^X\right) ^n\longrightarrow P^X\) given, for each \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\) and \(x\in X\), by

$$\begin{aligned} \overline{{{\mathscr{A}}}}\left( \overrightarrow{\zeta }\right) (x) ={{\mathscr{A}}}_x\left( \zeta _1(x),\zeta _2(x),\ldots ,\zeta _n(x)\right) \end{aligned}$$
(1)

is an nAAO on \(P^X\).

Proof

It is completed via below two steps.

First, for any \(x\in X\), from items (\({{\mathscr{A}}}\)1) and (\({{\mathscr{A}}}\)2) of Definition 2.1, one has that

$$\begin{aligned} \overline{{{\mathscr{A}}}}\left( \overrightarrow{\bot _{0_P}}^I\right) (x)&={{\mathscr{A}}}_x\left( \bot _{0_P}(x),\bot _{0_P}(x),\ldots ,\bot _{0_P}(x)\right) \\&={{\mathscr{A}}}_x\left( 0_P,0_P,\ldots ,0_P\right) \\&={{\mathscr{A}}}_x\left( \overrightarrow{0_P}^{I}\right) \\&=0_P \\&=\bot _{0_P}(x) \end{aligned}$$

and

$$\begin{aligned} \overline{{{\mathscr{A}}}}\left( \overrightarrow{\top _{1_P}}^I\right) (x)&={{\mathscr{A}}}_x\left( \top _{1_P}(x),\top _{1_P}(x),\ldots ,\top _{1_P}(x)\right) \\&={{\mathscr{A}}}_x\left( 1_P,1_P,\ldots ,1_P\right) \\&={{\mathscr{A}}}_x\left( \overrightarrow{1_P}^{I}\right) \\&=1_P \\&=\top _{1_P}(x). \end{aligned}$$

Therefore, one concludes that \(\overline{{{\mathscr{A}}}} \left( \overrightarrow{\bot _{0_P}}^I\right) =\bot _{0_P}\) and \(\overline{{{\mathscr{A}}}}\left( \overrightarrow{\top _{1_P}}^I\right) =\top _{1_P}\).

Second, for any \(\overrightarrow{\zeta },\overrightarrow{\eta }\in \left( P^X\right) ^n\) with \(\overrightarrow{\zeta }\precapprox \overrightarrow{\eta }\), it holds that \(\zeta _i\preceq \eta _i\) \((i=1,2,\ldots ,n)\), that is, for any \(x\in X\), it holds that

$$\begin{aligned} \zeta _i(x)\le _P\eta _i(x)~(i=1,2,\ldots ,n). \end{aligned}$$

Furthermore, for any \(x\in X\), from (\({{\mathscr{A}}}\)3) of Definition 2.1, one has that

$$\begin{aligned} \overline{{{\mathscr{A}}}}\left( \overrightarrow{\zeta }\right) (x)&={{\mathscr{A}}}_x\left( \zeta _1(x),\zeta _2(x),\ldots ,\zeta _n(x)\right) \\&\le _P{{\mathscr{A}}}_x\left( \eta _1(x),\eta _2(x),\ldots ,\eta _n(x)\right) \\&=\overline{{{\mathscr{A}}}}\left( \overrightarrow{\eta }\right) (x). \end{aligned}$$

Therefore, one concludes that \(\overline{{{\mathscr{A}}}}\left( \overrightarrow{\zeta }\right) \preceq \overline{{{\mathscr{A}}}}\left( \overrightarrow{\eta }\right)\).

Remark 3.1

Proposition 3.1 states ordinaria form to obtain nAAOs on function space \(P^X\) from known nAAOs on bounded poset P. In addition, it also states a unified manner of constructing the commonly used AOs on function spaces via a family of known ones.

Definition 3.1

An nAAO \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\) on \(P^X\) is called representable if there exists a family of nAAOs \(\left\{ {{\mathscr{A}}}_x:P^n\longrightarrow P\mid x\in X\right\}\) such that \({\text{A}}\) equals to \(\overline{{{\mathscr{A}}}}\) given in Eq. (1).

Theorem 3.1

Let \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\)  be an nAAO on \(P^X\). Then below are equivalent:

  1. (i)

    \({\text{A}}\) is representable;

  2. (ii)

    \({\text{A}}\left( {\vec{\zeta }} \right)\left( {x^{\dag } } \right) = {\text{A}}\left( {\vec{\eta }} \right)\left( {x^{\dag } } \right)\) if there are \(\overrightarrow{\zeta },\overrightarrow{\eta }\in \left( P^X\right) ^n\) satisfying \(\zeta _i\left( x^\dag \right) =\eta _i\left( x^\dag \right)\) \((i=1,2,\ldots ,n)\) for \(x^\dag \in X\).

Proof

  • (i) implies (ii): By Definition 3.1, it holds that there exists a family of nAAOs \(\left\{ {{\mathscr{A}}}_x:P^n\longrightarrow P\mid x\in X\right\}\) satisfying, for any \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\) and \(x\in X\),

    $${\text{A}}\,\left( {\mathop \zeta \limits^{ \to } } \right)\,\left( x \right)\, = \,\overline{{{\mathscr{A}}}}\,\left( {\mathop \zeta \limits^{ \to } } \right)\,\left( x \right)\, = \, {{\mathscr{A}}}_x \,\left( {\zeta _{1} \,\left( x \right),\,\zeta _{2} \,\left( x \right), \ldots ,\zeta _{n} \,\left( x \right)} \right)$$

    Furthermore, if there have \(\overrightarrow{\zeta },\overrightarrow{\eta }\in \left( P^X\right) ^n\) satisfying \(\zeta _i\left( x^\dag \right) =\eta _i\left( x^\dag \right)\) \((i=1,2,\ldots ,n)\) for \(x^\dag \in X\), then, it holds that

    $$\begin{aligned}{\text{A}}\left( {\vec{\zeta }} \right)\left( {x^{{\dag}} } \right)\; &= {{\mathscr{A}}}_{{x^{{\dag}} }} \left( {\zeta _{1} \left( {x^{{\dag}} } \right),\zeta _{2} \left( {x^{{\dag}} } \right), \ldots ,\zeta _{n} \left( {x^{{\dag}} } \right)} \right)\;\\& = {{\mathscr{A}}}_{{x^{{\dag}} }} \left( {\eta _{1} \left( {x^{{\dag}} } \right),\eta _{2} \left( {x^{{\dag}} } \right), \ldots ,\eta _{n} \left( {x^{{\dag}} } \right)} \right)\;\\& = {\text{A}}\left( {\vec{\eta }} \right)\left( {x^{{\dag}} } \right).\end{aligned}$$

  • (ii) implies (i): For every \(x\in X\), put operator \({{\mathscr{A}}}_x:P^n\longrightarrow P\) as

    $${{\mathscr{A}}}_x \left( {\vec{a}} \right) = {\text{A}}\left( {\vec{a}_{X} } \right)(x)$$

    for any \(\overrightarrow{a}\in P^n\). Then \({{\mathscr{A}}}_x\) is well-defined and the proof can be completed as follows.

  • Step 1. For each \(x\in X\), \({{\mathscr{A}}}_x\) is an nAAO on P.

    For each \(x\in X\), items (\({{\mathscr{A}}}\)1)–(\({{\mathscr{A}}}\)3) of Definition 2.1 can be proven as follows.

(\({{\mathscr{A}}}\)1):

For every \(x\in X\), it holds that

$$\begin{aligned}{{\mathscr{A}}}_x \left( {\overrightarrow {{0_{P} }} ^{I} } \right) &= {\text{A}}\left( {\left( {\overrightarrow {{0_{P} }} ^{I} } \right)_{X} } \right)(x) \\&= {\text{A}}\left( {(0_{P} )_{X} ,(0_{P} )_{X} , \ldots ,(0_{P} )_{X} } \right)(x) \\&= {\text{A}}\left( { \bot _{{0_{P} }} , \bot _{{0_{P} }} , \ldots , \bot _{{0_{P} }} } \right)(x) \\&={\text{A}}\left( {\overrightarrow {{ \bot _{{0_{P} }} }} ^{I} } \right)(x) \\&= \bot _{{0_{P} }} (x) \\&= 0_{P}.\end{aligned}$$

Accordingly, for each \(x\in X\), \({{\mathscr{A}}}_x\) satisfies them (\({{\mathscr{A}}}\)1) of Definition 2.1.

\(({{\mathscr{A}}}2)\):

For arbitrary \(x\in X\), one has that

$$\begin{aligned}{{\mathscr{A}}}_x \left( {\overrightarrow {{1_{P} }} ^{I} } \right) &= {\text{A}}\left( {\left( {\overrightarrow {{1_{P} }} ^{I} } \right)_{X} } \right)(x) \\&= {\text{A}}\left( {(1_{P} )_{X} ,(1_{P} )_{X} , \ldots ,(1_{P} )_{X} } \right)(x) \\&= {\text{A}}\left( { \top _{{1_{P} }} , \top _{{1_{P} }} , \ldots , \top _{{1_{P} }} } \right)(x) \\&={\text{A}}\left( {\overrightarrow {{ \top _{{1_{P} }} }} ^{I} } \right)(x) \\&= \top _{{1_{P} }} (x) \\&= 1_{P}.\end{aligned}$$

Accordingly, for each \(x\in X\), \({{\mathscr{A}}}_x\) satisfies item (\({{\mathscr{A}}}\)2) of Definition 2.1.

(\({{\mathscr{A}}}\)3):

For arbitrary \(\overrightarrow{a},\overrightarrow{b}\in P^n\) with \(\overrightarrow{a}\lessdot \overrightarrow{b}\), it holds that \(a_i\le _P b_i~(i=1,2,\ldots ,n)\). And thus, it holds that \((a_i)_X\preceq (b_i)_X~(i=1,2,\ldots ,n)\), that is, \(\overrightarrow{a}_X\lessapprox \overrightarrow{b}_X\). Furthermore, for any \(x\in X\), one has that

$${{\mathscr{A}}}_x \left( {\vec{a}} \right) = {\text{A}}\left( {\vec{a}_{X} } \right)(x) \le _{P} {\text{A}}\left( {\vec{b}_{X} } \right)(x) = {{\mathscr{A}}}_x \left( {\vec{b}} \right).$$

Therefore, for each \(x\in X\), \({{\mathscr{A}}}_x\) satisfies item (\({{\mathscr{A}}}\)3) of Definition 2.1.

  • Step 2. For any \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\) and \(x\in X\), \({\text{A}}\left( {\vec{\zeta }} \right)(x) = {{\mathscr{A}}}_{x} \left( {\zeta _{1} (x),\zeta _{2} (x), \ldots ,\zeta _{n} (x)} \right)\).

    Actually, for any \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\) and \(x\in X\), one has that

    $$\begin{aligned}{\text{A}}\left( {\vec{\zeta }} \right)(x) &= {\text{A}}\left( {\zeta _{1} ,\zeta _{2} , \ldots ,\zeta _{n} } \right)(x) \\&={\text{A}}\left( {(\zeta _{1} (x))_{X} ,(\zeta _{2} (x))_{X} , \ldots ,(\zeta _{n} (x))_{X} } \right)(x) \\&= {{\mathscr{A}}}_{x} (\zeta _{1} (x),\zeta _{2} (x), \ldots ,\zeta _{n} (x)).\end{aligned}$$

Proposition 3.2

If an nAAO \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\)  on \(P^X\) is representable, then the representation is unique.

Proof

Suppose there are two families of nAAOs \(\left\{ {{\mathscr{A}}}_x:P^n\longrightarrow P\mid x\in X\right\}\) and \(\left\{ {{\mathscr{B}}}_x:P^n\longrightarrow P\mid x\in X\right\}\) satisfying, for any \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\) and \(x\in X\),

$${\text{A}}\left( {\vec{\zeta }} \right)(x) = {{\mathscr{A}}}_x \left( {\zeta _{1} (x),\zeta _{2} (x), \ldots ,\zeta _{n} (x)} \right)$$

and

$${\text{A}}\left( {\vec{\zeta }} \right)(x) = {{\mathscr{B}}}_{x} \left( {\zeta _{1} (x),\zeta _{2} (x), \ldots ,\zeta _{n} (x)} \right).$$

Then, for each \(x\in X\) and every \(\overrightarrow{a}\in P^n\), it holds that

$$\begin{aligned}{{\mathscr{A}}}_{x} \left( {\vec{a}} \right) &= {{\mathscr{A}}}_{x} \left( {a_{1} ,a_{2} , \ldots ,a_{n} } \right) \\&= {{\mathscr{A}}}_{x} \left( {(a_{1} )_{X} (x),(a_{2} )_{X} (x), \ldots ,(a_{n} )_{X} (x)} \right) \\&= {\text{A}}\left( {\vec{a}_{X} } \right)(x) \\&= {{\mathscr{B}}}_{x} \left( {(a_{1} )_{X} (x),(a_{2} )_{X} (x), \ldots ,(a_{n} )_{X} (x)} \right)\\& = {{\mathscr{B}}}_{x} \left( {a_{1} ,a_{2} , \ldots ,a_{n} } \right) \\&= {{\mathscr{B}}}_{x} \left( {\vec{a}} \right).\end{aligned}$$

Furthermore, one get that \({{\mathscr{A}}}_x={{\mathscr{B}}}_x\) for any \(x\in X\). And thus, the representation of \({\text{A}}\) is unique.  

4 Fundamental properties of representable nAAOs on function spaces

Herein, we consider elementary properties of representable nAAOs on \(P^X\), which include idempotency, symmetry, associativity and bisymmetry.

Proposition 4.1

Let \({\text{A}}:\left( {P^{X} } \right)^{n}\longrightarrow P^{X}\)  be a representable nAAO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) is idempotent;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P^n \longrightarrow P\) is idempotent.

Proof

  • (i) implies (ii): For each \(x\in X\) and any \(a\in P\), it holds that

    $$\begin{aligned}{{\mathscr{A}}}_{x} \left( {\vec{a}^{I} } \right) &={{\mathscr{A}}}_{x} \left( {a,a, \ldots ,a} \right) \\&={{\mathscr{A}}}_{x} \left( {a_{X} (x),a_{X} (x), \ldots ,a_{X} (x)} \right)\\& = {\text{A}}\left( {\overrightarrow {{a_{X} }} ^{I} } \right)(x) \\&= {\text{A}}\left( {a_{X} ,a_{X} , \ldots ,a_{X} } \right)(x)\\& = a_{X} (x)\\& = a.\end{aligned}$$

    Thus, one has that, for each \(x\in X\), \({{\mathscr{A}}}_x:P^n\longrightarrow P\) is idempotent.

  • (ii) implies (i): For each \(\zeta \in P^X\) and every \(x\in X\), one has that

    $$\begin{aligned}{\text{A}}\left( {\vec{\zeta }^{I} } \right)(x) &= {{\mathscr{A}}}_{x} \left( {\zeta (x),\zeta (x), \ldots ,\zeta (x)} \right) \\&= \zeta (x),\end{aligned}$$

    that is, \({\text{A}}\left( {\vec{\zeta }^{I} } \right) = \zeta\). And thus, \({\text{A}}\) is idempotent.

\(\square\)

Proposition 4.2

Let \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\)  be a representable nAAO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) is symmetric;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P^n \longrightarrow P\) is symmetric.

Proof

  • (i) implies (ii): For each \(x\in X\), any \(\overrightarrow{a}\in P^n\) and all permutations \((p(1),p(2),\ldots ,p(n))\) of \((1,2,\ldots ,n)\), one has that

    $$\begin{aligned}{{\mathscr{A}}}_{x} \left( {\vec{a}} \right) &= {{\mathscr{A}}}_{x} \left( {(a_{1} )_{X} (x),(a_{2} )_{X} (x), \ldots ,(a_{n} )_{X} (x)} \right)\\& = {\text{A}}\left( {\vec{a}_{X} } \right)(x)\\& = {\text{A}}\left( {(a_{1} )_{X} ,(a_{2} )_{X} , \ldots ,(a_{n} )_{X} } \right)(x)\\& = {\text{A}}\left( {\left( {a_{{p(1)}} } \right)_{X} ,\left( {a_{{p(2)}} } \right)_{X} , \ldots ,\left( {a_{{p(n)}} } \right)_{X} } \right)(x)\\& = {{\mathscr{A}}}_{x} \left( {\left( {a_{{p(1)}} } \right)_{X} (x),\left( {a_{{p(2)}} } \right)_{X} (x), \ldots ,\left( {a_{{p(n)}} } \right)_{X} (x)} \right) \\&= {{\mathscr{A}}}_{x} \left( {a_{{p(1)}} ,a_{{p(2)}} , \ldots ,a_{{p(n)}} } \right).\end{aligned}$$

    Thus, for arbitrary \(x\in X\), \({{\mathscr{A}}}_x:P^n\longrightarrow P\) is symmetric.

  • (ii) implies (i): For each \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\), any \(x\in X\) and all permutations \((p(1),p(2),\ldots ,p(n))\) of \((1,2,\ldots ,n)\), it holds that

    $$\begin{aligned}{\text{A}}\left( {\vec{\zeta }} \right)(x)&= {{\mathscr{A}}}_{x} \left( {\zeta _{1} (x),\zeta _{2} (x), \ldots ,\zeta _{n} (x)} \right) \\&= {{\mathscr{A}}}_{x} \left( {\zeta _{{p(1)}} (x),\zeta _{{p(2)}} (x), \ldots ,\zeta _{{p(n)}} (x)} \right) \\&={\text{A}}\left( {\zeta _{{p(1)}} ,\zeta _{{p(2)}} , \ldots ,\zeta _{{p(n)}} } \right)(x),\end{aligned}$$

    that is, \({\text{A}}\left( {\vec{\zeta }} \right) = {\text{A}}\left( {\zeta _{{p(1)}} ,\zeta _{{p(2)}} , \ldots ,\zeta _{{p(n)}} } \right)\). And thus, it holds that \({\text{A}}\) is symmetric.

\(\square\)

Proposition 4.3

Let \({\text{A}}:\left( {P^{X} } \right)^{n} \longrightarrow P^{X}\) be a representable nAAO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) has \(\alpha\) as the annihilator;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P^n\longrightarrow P\) has \(\alpha (x)\) as the annihilator.

Proof

  • (i) implies (ii): For each \(x\in X\) and any \(\overrightarrow{a}\in P^n\), if there exists \(i'\in \{1,2,\ldots ,n\}\) satisfying \(a_{i'}=\alpha (x)\), then, it holds that

    $$\begin{aligned}{{\mathscr{A}}}_{x} \left( {a_{1} , \ldots ,a_{{i^{\prime}}} , \ldots ,a_{n} } \right) &={{\mathscr{A}}}_{x} \left( {(a_{1} )_{X} (x), \ldots ,\alpha (x), \ldots ,(a_{n} )_{X} (x)} \right) \\&= {\text{A}}\left( {(a_{1} )_{X} , \ldots ,\alpha , \ldots ,(a_{n} )_{X} } \right)(x)\\& = \alpha (x).\end{aligned}$$

    Thus, one has that, for each \(x\in X\), \({{\mathscr{A}}}_x:P^n\longrightarrow P\) has \(\alpha (x)\) as the annihilator.

  • (ii) implies (i): For each \(\overrightarrow{\zeta }\in \left( P^X\right) ^n\), if there exists \(i^*\in \{1,2,\ldots ,n\}\) satisfying \(\zeta _{i^*}=\alpha\), then, for any \(x\in X\), it holds that

    $$\begin{aligned}{\text{A}}\left( {\vec{\zeta }} \right)(x) &= {{\mathscr{A}}}_{x} \left( {\zeta _{1} (x), \ldots ,\zeta _{{i^{*} }} (x), \ldots ,\zeta _{n} (x)} \right) \\&= {{\mathscr{A}}}_{x} \left( {\zeta _{1} (x), \ldots ,\alpha (x), \ldots ,\zeta _{n} (x)} \right) \\&= \alpha (x),\end{aligned}$$

    that is, \({\text{A}}\left( {\vec{\zeta }} \right) = \alpha .\) And thus, one concludes that \({\text{A}}\) has \(\alpha\) as the annihilator.

\(\square\)

Definition 4.1

Binary operator \({{\mathscr{B}}}:S\times S\longrightarrow S\) on any bounded poset S satisfies

  • \(1_S\)-section inflation if \({{\mathscr{B}}}(1_S,s)\ge _S s\) for any \(s\in S\);

  • \(1_S\)-section deflation if \({{\mathscr{B}}}(1_S,s)\le _S s\) for any \(s\in S\);

  • associativity if \({{\mathscr{B}}}\left( u,{{\mathscr{B}}}\left( v,w\right) \right) ={{\mathscr{B}}}\left( {{\mathscr{B}}}\left( u,v\right) ,w\right)\) for any \(u,v,w\in S\);

  • bisymmetry if \({{\mathscr{B}}}\left( {{\mathscr{B}}}\left( u,v\right) ,{{\mathscr{B}}}\left( u',v'\right) \right) ={{\mathscr{B}}}\left( {{\mathscr{B}}}\left( u,u'\right) ,{{\mathscr{B}}}\left( v,v'\right) \right)\) for arbitrary \(u,v,u',v'\in S\).

\(1_S\) is the neutral element of \({{\mathscr{B}}}\) iff \({{\mathscr{B}}}\) satisfies \(1_S\)-section inflation and \(1_S\)-section deflation simultaneously.

Proposition 4.4

Let \({\text{A}}:P^{X} \times P^{X} \longrightarrow P^{X}\) be a representable binary AO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) satisfies \(\top _{1_P}\)-section inflation;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) satisfies \(1_P\)-section inflation.

Proof

  • (i) implies (ii): For each \(x\in X\) and any \(a\in P\), it holds that

    $$\begin{aligned} {{\mathscr{A}}}_{x} (1_{P} ,a) & ={{\mathscr{A}}}_{x} ({ \top }_{{1_{P} }} (x),a_{X} (x)) \\& = {\text{A}}({ \top }_{{1_{P} }} ,a_{X} )(x) \\& \ge _{P} a_{X} (x) \\& = a. \end{aligned}$$

    Thus, for each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) satisfies \(1_P\)-section inflation.

  • (ii) implies (i): For each \(\alpha \in P^X\) and any \(x\in X\), one has that

    $$\begin{aligned}{\text{A}}({ \top }_{{1_{P} }} ,\alpha )(x) &= {{\mathscr{A}}}_{x} ({ \top }_{{1_{P} }} (x),\alpha (x))\\& = {{\mathscr{A}}}_{x} (1_{P} ,\alpha (x)) \\&\ge _{P} \alpha (x).\end{aligned}$$

    Furthermore, for each \(\alpha \in P^X\), it holds that \({\text{A}}({ \top }_{{1_{P} }} ,\alpha ){ \succeq }\alpha\), that is, \({\text{A}}\) satisfies \(\top _{1_P}\)-section inflation.

Proposition 4.5

Let \({\text{A}}:P^{X} \times P^{X} \longrightarrow P^{X}\) be a representable binary AO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) satisfies \(\top _{1_P}\)-section deflation;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) satisfies \(1_P\)-section deflation.

Proof

The proof is similar to Proposition 4.4.

Via Propositions 4.4 and 4.5, one can conclude below conclusion.

Theorem 4.1

Let \({\text{A}}:P^{X} \times P^{X} \longrightarrow P^{X}\)  be a representable binary AO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) has \(\top _{1_P}\) as the neutral element;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) has \(1_P\) as the neutral element.

Proposition 4.6

Let \({\text{A}}:P^{X} \times P^{X}\longrightarrow P^{X}\) be a representable binary AO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) is associative;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) is associative.

Proof

  • (i) implies (ii): For each \(x\in X\) and any \(a,b,c\in P\), one has that

    $$\begin{aligned}{{\mathscr{A}}}_{x} \left( {{{\mathscr{A}}}_{x} (a,b),c} \right) &= {{{\mathscr{A}}}}_{x} \left( {{{\mathscr{A}}}_{x} \left( {a_{X} (x),b_{X} (x)} \right),c_{X} (x)} \right) \\&= {{\mathscr{A}}}_{x} \left( {{\text{A}}\left( {a_{X} ,b_{X} } \right)(x),c_{X} (x)} \right) \\&= {\text{A}}\left( {{\text{A}}\left( {a_{X} ,b_{X} } \right),c_{X} } \right)(x) \\&= {\text{A}}\left( {a_{X} ,{\text{A}}\left( {b_{X} ,c_{X} } \right)} \right)(x)\\& = {{\mathscr{A}}}_{x} \left( {a_{X} (x),{\text{A}}\left( {b_{X} ,c_{X} } \right)(x)} \right)\\& = {{\mathscr{A}}}_{x} \left( {a_{X} (x),{{\mathscr{A}}}_{x} \left( {b_{X} (x),c_{X} (x)} \right)} \right)\\& = {{\mathscr{A}}}_{x} \left( {a,{{\mathscr{A}}}_{x} \left( {b,c} \right)} \right).\end{aligned}$$

    Thus, for each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) is associative.

  • (ii) implies (i): It is similar to the proof for associativity part of Proposition 2 in (Lobillo et al. 2021).

Proposition 4.7

Let \({\text{A}}:P^{X} \times P^{X} \longrightarrow P^{X}\) be a representable binary AO on \(P^X\). Then the following statements are equivalent:

  1. (i)

    \({\text{A}}\) is bisymmetric;

  2. (ii)

    For each \(x\in X\), \({{\mathscr{A}}}_x:P\times P \longrightarrow P\) is bisymmetric.

Proof

  • (i) implies (ii): For each \(x\in X\) and any \(a,b,c,d\in P\), it holds that

    $$\begin{aligned}{{\mathscr{A}}}_{x} \left( {{{\mathscr{A}}}_{x}\left( a ,b \right),{{\mathscr{A}}}_{x}\left( c ,d \right)} \right) &= {{\mathscr{A}}}_{x}\left( {{\mathscr{A}}}_{x}\left( {a_{X}(x),b_{X}(x)} \right),{{\mathscr{A}}}_{x}\left( {c_{X}(x),d_{X}(x)} \right) \right)\\& ={{\mathscr{A}}}_{x}\left( {{\text{A}}\left( {a_{X} ,b_{X} } \right)(x),{\text{A}}\left( {c_{X} ,d_{X} } \right)(x)} \right)\\&= {\text{A}}\left( {{\text{A}}\left( {a_{X} ,b_{X} } \right),{\text{A}}\left( {c_{X} ,d_{X} } \right)} \right)(x)\\&= {\text{A}}\left( {{\text{A}}\left( {a_{X} ,c_{X} } \right),{\text{A}}\left( {b_{X} ,d_{X} } \right)} \right)(x)\\& = {{\mathscr{A}}}_{x} \left( {{\text{A}}\left( {a_{X} ,c_{X} } \right)(x),{\text{A}}\left( {b_{X} ,d_{X} } \right)(x)} \right) \\&= {{\mathscr{A}}}_{x} \left( {{{\mathscr{A}}}_{x} \left( {a_{X} (x),c_{X} (x)} \right),{{\mathscr{A}}}_{x} \left( {b_{X} (x),d_{X} (x)} \right)} \right) \\&= {{\mathscr{A}}}_{x} \left( {{{\mathscr{A}}}_{x} (a,c),{{\mathscr{A}}}_{x} \left( {b,d} \right)} \right).\end{aligned}$$

    Thus, for each \(x\in X\), \({{\mathscr{A}}}_x:P\times P\longrightarrow P\) is bisymmetric.

  • (ii) implies (i): For each \(\alpha ,\beta ,\gamma ,\delta \in P^X\) and any \(x\in X\), it holds that

    $$\begin{aligned}{\text{A}}\left( {\text{A}\left( {\alpha ,\beta } \right),{\text{A}}\left( {\gamma ,\delta } \right)} \right)(x) &= {{\mathscr{A}}}_{x} \left( {\text{A}\left( {\alpha ,\beta } \right)(x),{\text{A}}\left( {\gamma ,\delta } \right)(x)} \right)\\& = {{\mathscr{A}}}_{x} \left( {{{\mathscr{A}}}_{x} \left( {\alpha (x),\beta (x)} \right),{{\mathscr{A}}}_{x} \left( {\gamma (x),\delta (x)} \right)} \right)\\& = {{\mathscr{A}}}_{x} \left( {{\mathscr{A}}}_{x} \left( {\alpha (x),\gamma (x)} \right),{{\mathscr{A}}}_{x} \left( {\beta (x),\delta (x)} \right) \right) \\&= {{\mathscr{A}}}_{x} \left( {{\text{A}}\left( {\alpha ,\gamma } \right)(x),{\text{A}}\left( {\beta ,\delta } \right)(x)} \right) \\&={\text{A}}\left( {\text{A}\left( {\alpha ,\gamma } \right),\text{A}\left( {\beta ,\delta } \right)} \right)(x).\end{aligned}$$

    Furthermore, for each \(\alpha ,\beta ,\gamma ,\delta \in P^X\), \({\text{A}}\left( {{\text{A}}\left( {\alpha ,\beta } \right),{\text{A}}\left( {\gamma ,\delta } \right)} \right) ={\text{A}}\left( {{\text{A}}\left( {\alpha ,\gamma } \right),{\text{A}}\left( {\beta ,\delta } \right)} \right)\), that is, \({\text{A}}\) is bisymmetric.

5 Conclusion

The study focuses on construction method of nAAOs on function spaces composed of all fuzzy sets with bounded posets as truth values set along with the basic properties of such obtained nAAOs on function spaces.

Major results are as follows.

  • For arbitrary bounded poset P, based on a family of nAAOs on P, we gain the construction method of nAAOs on function space \(P^X\) composed of all P-fuzzy subsets of X with P as the truth values set.

  • After introducing the concept of representable nAAOs on \(P^X\), their equivalent characterization is obtained.

  • Basic properties of representable nAAOs on \(P^X\) are discussed, which contain idempotency, symmetry, associativity and bisymmetry.

The following work can be considered are: (1) the study of other properties of representable nAAOs on function spaces beside those discussed in this paper; (2) the other construction methods for nAAOs on function spaces; (3) the practical applications of the theoretical results obtained for representable nAAOs on function spaces cause of that it covers the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets and those two classes of extended fuzzy sets can describe more uncertainty and better deal with practical application issues involving in decision-making, image processing, artificial neural network, clustering, fuzzy inference, vehicle guidance system, edge detection, and so on.