P-Algebras

Abstract

Following the notions of $\mathrm{\Omega }$-set and $\mathrm{\Omega }$-algebra where $\mathrm{\Omega }$ is a complete lattice, we introduce P-algebras, replacing the lattice $\mathrm{\Omega }$ by a poset P. A P-algebra is a classical algebraic structure in which the usual equality is replaced by a P-valued equivalence relation, i.e., with the symmetric and transitive map from the underlying set into a poset P. In addition, this generalized equality is (as a map) compatible with the fundamental operations of the algebra. The diagonal restriction of this map is a P-valued support of a P-algebra. The particular subsets of this support, its cuts, are classical subalgebras, while the cuts of the P-valued equality are congruences on the corresponding cut subalgebras. We prove that the collection of the corresponding quotients of these cuts is a centralized system in the lattice of weak congruences of the basic algebra. We also describe the canonical representation of P-algebras, independent of the poset P.

1. Introduction

This paper originates in the field of generalized characteristic functions of sets and relations, providing functions that extend the notion of a subset, or a subrelation. The co-domain of these functions is usually an ordered structure. In this context, the Fuzzy set theory is well known and one of its main tools in the corresponding investigations are so-called cut sets and relations (see, e.g., [1]), which we also use here. The cuts are particular subsets (subrelations) of the function’s domain, determined by the value from the co-domain lattice. Another independently developed theory dealing with generalizations of characteristic functions is omega sets, introduced by Fourman and Scott in [2]. They were considering a set equipped with the particular generalized equality, being a symmetric and transitive map from this set into a Heyting lattice. Their aim was to model axioms of the Set Theory, and in this way, they obtained suitable objects (functions) acting as sets. Let us mention that the kind of generalized equality as a function from the square of the domain to the co-domain lattice was also investigated by Höhle ([3,4]), Demirci ([5,6,7]), Belohlavek, Vychodil ([8,9]) and others within the theory of fuzzy sets. Our goal when we started our investigations, which continue in the present paper, was to introduce an extension of omega sets by considering algebras instead of sets only. In this way, we obtained omega algebras.

In particular, we were dealing with omega algebras as general structures [10], defining how identities hold in them as lattice formulas. We proved that for an omega algebra, all quotients of cut subalgebras over the corresponding cut congruences are classical algebras of the same type, fulfilling the corresponding identities. Then, we dealt with omega quasigroups together with a new way of approximating the solution of linear equations [11] and with some other special algebraic structures. In our further research, we extended the co-domain ordered structure, which was a lattice. In paper [12], we introduced the notion of a poset-valued set, also named a P-set. A P-set is a pair consisting of a set and a generalized equality relation as a special map from $A^2$ to P, extending the classical equality relation. In the mentioned paper, we proved that a P-set is characterized by a centralized system in the family of all weak equivalences in the underlying set A. Consequently, the canonical representation of a P-set is defined using the mentioned centralized system as the related poset.

As the next step, in the present research, we naturally use the newly introduced concept of P-sets to introduce and investigate the related poset-valued algebras (P-algebras).

Our manuscript aims to investigate the concept of P-algebras for general algebraic structures. Therefore, all the general results are valid for specialized algebraic structures, as P-groups, P-rings, P-lattices, and others. Such specialized structures are not defined, but the definitions can be easily deduced from the general definition of P-algebras. We prove that cuts of the diagonal subrelation of the P-valued equality for a P-algebra are subalgebras of its basic algebra. Similarly, cuts of the P-valued equality itself are congruences on the corresponding subalgebras—i.e., weak congruences on the basic algebra. Consequently, the collection of all the obtained quotient structures is a centralized system in the lattice of weak congruences of this algebra. Further, if ordered dually to the set inclusion, this collection is used for the construction of the canonical representation of this P-algebra, in which the poset P is replaced by the mentioned poset of cut-quotients. In the framework of set-theoretic language, we also define how identities hold on P-algebras. It turns out that in this case, the cut-quotients classically satisfy these identities. As for omega algebras, this is a necessary and sufficient condition under which a P-algebra satisfies identities in the language of set-theoretic formulas.

Our results are illustrated by suitable examples.

Let us finally mention that by generalizing lattice-valued algebraic structures to poset-valued ones, we fully change the syntax of our framework. While in the lattice case, i.e., for $\mathrm{\Omega }$-algebras, we use lattice-theoretic formulas, in the poset framework i.e., for P-algebras, we strictly apply the set-theoretic syntax; we use it for introducing and investigating relevant properties of the new structures.

2. Preliminaries

We list here some notions that we use. These are connected to sets, families of particular subsets, relations (like equivalence and order) and functions. We also recall some basic features from the theory of ordered sets and general algebra.

If X is a nonempty set, then a collection of subsets $\mathcal{C}$ closed under all intersections is called a closure system. A generalization of this notion is a collection of subsets $\mathcal{C}$ closed under centralized intersections, meaning that for all $x\in X$, $\bigcap \{Y\in \mathcal{C}\mid x\in Y\}\in \mathcal{C}.$ Such collection of subsets is called a centralized system.

A partially ordered set (poset) is a set with an ordering relation (reflexive, antisymmetric and transitive), denoted by $P:=(P,⩽)$. We also use other related relations on P, ⩾, < and > defined by the following:

$x⩾y$ if and only if $y⩽x$;

$x<y$ if and only if $x⩽y$ and $x\ne y$;

$x>y$ if and only if $y⩽x$ and $x\ne y$.

We mention the important properties and subsets of a poset P that we use.

The poset $(P,⩽)$ may have the greatest element (also called the top element, or the top), denoted by 1:

for every $p\in P$, $p⩽1$.

Clearly, the top element, if it exists, is unique.

Analogously, the poset may have the smallest element (the bottom), denoted by 0:

for every $p\in P$, $0⩽p$.

As for the top element, if the bottom exists, it is unique.

The principal ideal generated with $p\in P$ is defined with $\downarrow p=\{x\in P\mid x⩽p\}$ and the dual notion is the principal filter generated by p: $\uparrow p=\{x\in P\mid p⩽x\}$.

If $(P,⩽)$ and $(Q,⩽)$ are posets, then the bijective map $f:P\to Q$ that fulfills

$$x⩽y\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}f\left(x\right)⩽f\left(y\right)$$

is an order-isomorphism.

More details and further results from the theory of ordered sets and lattice theory can be found e.g., in books [13,14].

If A is a nonempty set, then a symmetric and transitive relation E on A is a weak equivalence on A. Observe that a relation E on A with these properties can fulfill. Also, $(x,y)\in E$ implies $(x,x)\in E$ and $(y,y)\in E$. As a consequence, we have the following.

Lemma 1.

Every weak equivalence E on A is an equivalence relation on a subset $A_1$ of A where $A_1=\{x\in A\mid (x,x)\in E\}$.

The collection of all weak equivalences on a nonempty set A forms a complete lattice under the set inclusion.

If A is a nonempty set and P is a poset, then any map $\mu :A\to P$ is poset-valued map, or a P-map. In particular, if the domain of a poset-valued map $\rho$ is a square, i.e., if we have $\rho :A^2\to P$, then the map $\rho$ is a poset-valued (binary) relation on A. For a poset-valued map $\mu :A\to P$ and $p\in P$, we define the p-cut or simply cut the subset ${\mu }_p$ of A by ${\mu }_p:={\mu }^{-1}(\uparrow p)$. Obviously, ${\mu }_p=\{x\in A\mid \mu \left(x\right)⩾p\}$. By this definition, the cut ${\rho }_p$ of a poset-valued relation $\rho :A^2\to P$ is a classical binary relation on A: ${\rho }_p=\{(x,y)\mid \rho (x,y)⩾p\}$. The families of all cuts of a P-valued set $\mu$, or of a P-valued relation $\rho$, are denoted by the capital index, namely by ${\mu }_P$ and ${\rho }_P$, respectively: ${\mu }_P:=\{{\mu }_p\mid p\in P\}$ and ${\rho }_P:=\{{\rho }_p\mid p\in P\}$.

The following is a known basic property of cuts for P-valued functions (see e.g., [15]).

Proposition 1.

For any P-valued function $\mu :A\to P$, for every $a\in A$,

$$\mu \left(a\right)=\bigvee (p\mid a\in {\mu }_p);and\bigcap ({\mu }_p\mid a\in {\mu }_p)={\mu }_{\bigvee \left(p\right|a\in {\mu }_p)}.$$

A consequence of this proposition is that the collection of all cuts of a poset-valued set $\mu :A\to P$ is a centralized system over A.

For $\mu :A\to P$, let ∼ be a relation on a poset P, defined by

$$p\sim q\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\mu }_p={\mu }_q.$$

(1)

Relation ∼ is the kernel of the map $f:P\to \mathcal{P}\left(A\right)$ defined by $f\left(p\right)={\mu }_p$. Therefore, ∼ is an equivalence relation on P.

If $(P,⩽)$ is a poset, then a $\mathit{P}$-valued weak equivalence relation on a nonempty set A is a P-valued relation $E:A^2\to P$ on A, which for all $x,y\in A$ fulfills the symmetry (S) and transitivity (T), as follows:

(S): $E(x,y)=E(y,x)$;

(T): $\downarrow E(x,y)\bigcap \downarrow E(y,z)\subseteq \downarrow E(x,z)$.

A P-valued equality on A is a P-valued equivalence which additionally satisfies the separability condition: for all $x,y\in A$.

$E(x,y)\nless E(x,x)$ implies $x=y$ unless P has the bottom element 0 and $E(x,x)=0$.

Clearly, if for some $x\in A$, $E(x,x)=0$, then by S and T, for every y, we have $E(x,y)=0$ as well.

If $\mathcal{A}=(A,F)$ is an algebra, then a weak equivalence on A which is compatible with all the operations in F is a weak congruence relation on $\mathcal{A}$.

The following holds by Lemma 1 and by the definition of a weak congruence.

Lemma 2.

For every weak congruence ρ on $\mathcal{A}$, the set $B=\{x\in A\mid (x,x)\in \rho \}$ is a subalgebra of $\mathcal{A}$, and ρ is a classical congruence on B.

The set of all weak congruences on $\mathcal{A}$ forms an algebraic lattice under the set inclusion. This lattice is denoted by ${\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$.

Let $A\ne \varnothing$, let P be a poset and E a P-valued equality on A. Then, as defined in [12], a pair $(A,E)$ is called a $\mathit{P}$-set.

In this context, the P-valued map $\mu :A\to P$, defined with $\mu \left(x\right):=E(x,x)$ is called the support of E.

If P is a complete lattice (P is then usually denoted by $\mathrm{\Omega }$), then $(A,E)$ is an $\mathrm{\Omega }$-set [12].

Some basic properties of P-sets were elaborated in the paper [12]. We omit the proofs, they can be found in the mentioned paper, together with some other relevant properties and examples.

Theorem 1.

Let $(A,E)$ be a P-set with the support μ. The families of cuts $E_P=\{E_p\mid p\in P\}$ of E and ${\mu }_P=\{{\mu }_p\mid p\in P\}$ of μ are centralized systems and every $E_p$ is an equivalence relation on ${\mu }_p$.

According to the last theorem, for every $p\in P$, the quotient ${\mu }_p/E_p=\{{\left[x\right]}_{E_p}\mid x\in {\mu }_p\}$ is a partition of the subset ${\mu }_p$ of A induced by the equivalence relation $E_p$ over this subset. These quotient sets are the basic features of P-sets, establishing their link to classical set-theoretic structures. Moreover, not only that a P-set determine a collection of cut-quotient sets, but also the converse holds. This procedure which starts with a particular collection of weak equivalences enabling a construction of a P-set, is described in the sequel.

Theorem 2

([12]). Let $\mathcal{F}\subseteq Ew\left(A\right)$ be a centralized system over the set of weak equivalences of a nonempty set A, satisfying the condition

$$ifa\ne bthen(a,b)\notin \bigcap \{\rho \in \mathcal{F}\mid (a,a)\in \rho \}.$$

(2)

Then, $(A,E)$ is a P-set with

$$E(x,y)=\bigcap \{\rho \in \mathcal{F}\mid (x,y)\in \rho \},$$

where the poset $\mathcal{F}$ is considered under the order opposite to inclusion and $E_{\rho }=\rho$.

The P-set constructed in Theorem 2 from the set A using the centralized family of weak equivalences on A is said to be a canonical $\mathit{P}$-set. This name indicates the general role of this construction.

Let $P_1$ and $P_2$ be posets, let A be a set, and let $(A,E_1)$ and $(A,E_2)$ be P-sets.

$(A,E_1)$ and $(A,E_2)$ are cut-equivalent if their collections of cuts (factors) $\{{\mu }_p/E_p\mid p\in P\}$. coincide.

Theorem 3.

Let $(A,E)$ be a P-set where P is a poset. Then there is a unique canonical P-set which is cut-equivalent to $(A,E)$.

Let $(A,E)$ be a P-set and let $(A,E_1)$ be the unique canonical P-set which is cut-equivalent to $(A,E)$. We call the canonical P-set which is cut-equivalent to $(A,E)$, canonical representation of $(A,E)$.

Corollary 1.

Let A be a set. Let $P_1$ and $P_2$ be posets. Let $E_1:A\times A\to P_1$ and $E_2:A\times A\to P_2$ be poset-valued relations, such that $(A,E_1)$ and $(A,E_2)$ are P-sets. Then $(A,E_1)$ and $(A,E_2)$ are cut-equivalent if and only if they have the same canonical representation.

In the present research, we generalize $\mathrm{\Omega }$-algebras, introduced in [10].

3. Results: $\mathit{P}$-Algebra

Let $\mathcal{A}=(A,F)$ be an algebra and let $(P,⩽)$ be a poset. In case the algebra has a nullary operation, then P has the top element 1. We say that a P-valued function $\mu :A\to P$ is compatible with the operations on $\mathcal{A}$ if for every n-ary operation $f\in F$, $(n\ne 0)$, for all $a_1,\dots ,a_n\in A$ and for any constant $c\in F$,

$$\bigcap _{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow \mu \left(f(a_1,\dots ,a_n)\right)\mathrm{and}\mu \left(c\right)=1.$$

(3)

Analogously, a P-relation $\rho :A^2\to P$ on $\mathcal{A}$, is said to be compatible with the operation f, if for all $a_1,\dots ,a_n,b_1,\dots ,b_n\in A$ and for the constant c in F,

$$\bigcap _{i=1}^n\downarrow \rho (a_i,b_i)\subseteq \downarrow \rho (f(a_1,\dots ,a_n),f(b_1,\dots ,b_n))\mathrm{and}\rho (c,c)=1.$$

(4)

Observe that for $P=\left(\right\{0,1\},⩽)$, $\mu$ is a characteristic function of a subset of A and $\rho$ is a characteristic function of a binary relation R over A. In this case, Formula (3) expresses the compatibility of f over the subset M of A: $M=\{x\in A\mid \mu (x)=1\}$, and Formula (4) analogously present the compatibility of the binary relation R over A, whose characteristic function is $\rho$. As mentioned, if the algebra has nullary operations, then these operations have the greatest value of belonging to $\mu$ and they have the greatest value of relationship to themselves. Therefore, in the case of the existence of nullary operations, poset $(P,⩽)$ has to possess the greatest element 1.

Let $\mathcal{A}=(A,F)$ be an algebra and let $(P,⩽)$ be a poset. Let also $E:A^2\to P$ be a P-valued equality on A, which is compatible with the operations in $\mathcal{A}$. Then, the pair $(\mathcal{A},E)$ is a $\mathit{P}$-algebra. The classical algebra $\mathcal{A}$ is the underlying, basic algebra of $(\mathcal{A},E)$.

Theorem 4.

Let $(\mathcal{A},E)$ be a P-algebra. Then the following hold:

$\left(i\right)$ The P-valued function $\mu :A\to P$, determined by E with $\mu \left(x\right):=E(x,x)$ is compatible with the operations in F.

$\left(ii\right)$ For every $p\in P$, the cut ${\mu }_p$ is a subalgebra of $\mathcal{A}$, and $E_p$ is a congruence on ${\mu }_p$.

Proof. 

$\left(i\right)$ It is obvious that Equation (3) for $\mu \left(x\right)=E(x,x)$ and for each f, including the nullary operations, follows directly from Equation (4) with P-valued relation E.

$\left(ii\right)$ Let $p\in P$ and ${\mu }_p=\{x\in A\mid \mu \left(x\right)⩾p\}$. Then for every nullary operation $c\in F$, if there are nullary operations in F, we have $c\in {\mu }_p.$

Now, let $f\in F_n$ and let $a_1,...,a_n\in {\mu }_p$. Then, $\mu \left(a_i\right)⩾p$ and thus $p\in \downarrow \mu \left(a_i\right)$ for all $i\in \{1,\dots ,n\}.$ Hence, $p\in {\bigcap }_{i=1}^n\downarrow \mu \left(a_i\right)$ and by

${\bigcap }_{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow \mu \left(f(a_1,\dots ,a_n)\right)$

we have that $p\in \downarrow \mu \left(f(a_1,\dots ,a_n)\right)$ and $\mu \left(f(a_1,\dots ,a_n)\right)⩾p$, and hence $f(a_1,\dots ,a_n)\in {\mu }_p$. Therefore, ${\mu }_p$ is a subalgebra of $\mathcal{A}$.

$E_p$ is an equivalence relation on ${\mu }_p$ by Theorem 1. Now, we have to prove the compatibility.

Let $f\in F_n$ and let $x_1,...,x_n,y_1,...,y_n\in {\mu }_p$ such that $(x_i,y_i)\in E_p.$

This means that $p\in \downarrow E(x_i,y_i)$ for all i and hence

$p\in {\bigcap }_{i=1}^n\downarrow E(x_i,y_i)$ and by Formula (4),

$p\in \downarrow E(f(x_1,\dots ,x_n),f(y_1,\dots ,y_n))$ and thus $(f(x_1,\dots ,x_n),f(y_1,\dots ,y_n))\in E_p$. Therefore, $E_p$ is a congruence on ${\mu }_p$. □

Theorem 4 remains valid if P is a complete lattice, so this theorem is a generalization of the analogous theorem on $\mathrm{\Omega }$-algebras ([10]).

Theorem 5.

Let $\mathcal{A}=(A,F)$ be an algebra and let $(P,⩽)$ be a poset with the top element 1. Let $E:A^2\to P$ be a P-valued relation on A. Then, E is a P-valued compatible equivalence on $\mathcal{A}$ if and only if all the cut relations are weak congruences on $\mathcal{A}$.

Proof. 

From the fact that E is a P-valued weak equivalence relation on A compatible with the operations in $\mathcal{A}$, we have that all the cuts satisfy analogous conditions by Theorem 4. Now, suppose that $E:A^2\to P$ is a P-valued relation on A and that all the cut relations are weak equivalences on A compatible with operations on F. By Proposition 1, $E(x,y)=\bigvee (p\mid (x,y)\in E_p)$ and the supremum on the right exists in poset P. Since all the cuts are symmetric, it is easy to see that E is symmetric as well. To prove transitivity, suppose that $p\in \downarrow E(x,y)\bigcap \downarrow E(y,z)$. It follows that $p⩽E(x,y)$ and $p⩽E(y,z)$ and hence $(x,y),(y,z)\in E_p$. By the transitivity of $E_p$, $(x,z)\in E_p$ and $p\in \downarrow E(x,z)$.

To prove compatibility with an n-ary operation $f\in F$, suppose that $p\in {\bigcap }_{i=1}^n\downarrow E(a_i,b_i)$. It follows that $(a_i,b_i)\in E_p,$ for all $i\in \{1,...,n\}.$ By the compatibility of cut $E_p$, $f(a_1,\dots ,a_n),f(b_1,\dots ,b_n))\in E_p$ and hence $p\in \downarrow E(f(a_1,\dots ,a_n),f(b_1,\dots ,b_n))$. $E(c,c)=1$ for every constant $c\in F$ follows from the fact that for every $p\in P,$$(c,c)\in E_p$. □

We can note that in case there are no constants in F, the previous theorem is valid without the request that the poset P possesses the top element 1.

Proposition 2.

Let $(\mathcal{A},E)$ be a P-algebra and let $\mathcal{F}$ be a family of cuts of E. Then, for all $a,b\in A,$ such that $E(a,a)\ne 0$ if 0 exists in P,

$$if(a,b)\in \bigcap \{\rho \in \mathcal{F}\mid (a,a)\in \rho \}thena=b.$$

(5)

Proof. 

Let $E(a,a)\ne 0$ and suppose that $(a,b)\in \bigcap \{\rho \in \mathcal{F}\mid (a,a)\in \rho \}$. Since $(a,a)\in E_{E(a,a)}$, we have $\bigcap \{\rho \in \mathcal{F}\mid (a,a)\in \rho \}\subseteq E_{E(a,a)}$ and $(a,b)\in E_{E(a,a)}$ and hence $E(a,a)⩽E(a,b)$ and due to the separability, $a=b$. □

Lemma 3.

Let $(\mathcal{A},E)$ be a P-algebra. Let u be a term in the language of the algebra $\mathcal{A}$, whose variables are among $x_1,\dots ,x_n$. Then, for all $a_1,\dots ,a_n,b_1,\dots ,b_n\in A$,

$$\bigcap _{i=1}^n\downarrow E(a_i,b_i)\subseteq \downarrow E(u(a_1,\dots ,a_n),u(b_1,\dots ,b_n)).$$

(6)

Proof. 

The proof uses the induction on the complexity of the term u. Let k be the number of operational symbols in the term u. For $k=0$, u is a variable x and then the Equation (6) is trivially valid. If u is a term with one operational symbol f, then the Equation (6) is valid by (4).

Suppose that (6) is valid for all terms with less than k operational symbols and prove that it is true for a term with k operational symbols. Let $u=f(t_1,...,t_m)$ and let $x_1,...,x_n$ be all variables that appear in term u. Now for every $t_i$, for $i=1,...,m$,

$$\bigcap _{i=1}^n\downarrow E(a_i,b_i)\subseteq \downarrow E(t_i(a_1,\dots ,a_n),t_i(b_1,\dots ,b_n))$$

hence also

$$\bigcap _{i=1}^n\downarrow E(a_i,b_i)\subseteq \bigcap _{i=1}^m\downarrow E(t_i(a_1,\dots ,a_n),t_i(b_1,\dots ,b_n)).$$

By Formula (4),

$$\bigcap _{i=1}^m\downarrow E(t_i(a_1,...,a_n),t_i(b_1,...,b_n))\subseteq$$

$$\downarrow E(f(\left(t_1(a_1,\dots ,a_n)\right),...,t_m(a_1,...,a_m)),f(t_1(b_1,\dots ,b_n),...,t_m(b_1,...,b_n))).$$

By combining the last two formulas, we have the required proof. □

Corollary 2.

Let $(\mathcal{A},E)$ be a P-algebra. Let u be a term in the language of the algebra $\mathcal{A}$, whose variables are among $x_1,\dots ,x_n$. Let $\mu \left(x\right)=E(x,x)$. Then, for all $a_1,\dots ,a_n,b_1,\dots ,b_n\in A$,

$$\bigcap _{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow \mu \left(u(a_1,\dots ,a_n)\right).$$

(7)

Theorem 6.

Let $(\mathcal{A},E)$ be a P-algebra. Then, the collection $\{E_p\mid p\in P\}$ of cuts of E is a centralized system in the lattice ${\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$ of weak congruences of the basic algebra $\mathcal{A}$.

Proof. 

Straightforward by Proposition 1. □

Example 1.

Let $G=\{a,b,c,d\}$ and let the binary operation $\ast$ over G be defined by $x\ast y=x$. $\mathcal{G}=(G,\ast )$ is clearly a groupoid. Let the poset P be the one in Figure 1, and let the P-valued equality on G be given in

Table 1. P-valued relation E.

E	a	b	c	d
a	p	v	u	w
b	v	q	t	x
c	u	t	r	y
d	w	x	y	s

.

It is straightforward to check that E is symmetric and transitive. Moreover, the Formula (4) is obviously valid due to the simple definition of the operation *. In addition, the separability property is satisfied, since all the elements of the diagonal of E are strictly greater than the elements in the same row and column. Hence, $(\mathcal{G},E)$ is a P-algebra, i.e., a P-groupoid.

We can see that all the cuts of E are weak congruences on the groupoid $\mathcal{G}$, as follows:

$E_p=\left\{(a,a)\right\}$, $E_q=\left\{(b,b)\right\}$, $E_r=\left\{(c,c)\right\}$, $E_s=\left\{(d,d)\right\}$

$E_t=\{(a,a),(b,b),(c,c),(b,c),(c,b)\}$, $E_u=\{(a,a),(b,b),(c,c),(a,c),(c,a)\}$;

$E_v=\{(a,a),(b,b),(c,c),(a,b),(b,a)\}$, $E_w=\{(a,a),(d,d),(a,d),(d,a)\}$;

$E_x=\{(b,b),(d,d),(b,d),(d,b)\}$, $E_y=\{(c,c),(d,d),(c,d),(d,c)\}$.

The family of all cuts of E is a centralized system, which is easy to check.

Clearly, if P is a complete lattice, then the mentioned collection of cuts of E is a closure system in the weak congruence lattice of $\mathcal{A}$.

Let $u(x_1,\dots ,x_n)\approx v(x_1,\dots ,x_n)$, shortly $u\approx v$, be an identity in the language of the algebra $\mathcal{A}$, whose variables are among $x_1,\dots ,x_n$. Then, we say that the P-algebra $(\mathcal{A},E)$ satisfies this identity if for any $a_1,\dots ,a_n\in A$

$$\bigcap _{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n)),$$

(8)

where $\mu \left(a_i\right)=E(a_i,a_i)$.

As for the other properties, if P is a complete lattice, then the Formula (8) is equivalent to the lattice version.

It may happen that the basic algebra of a P-algebra fulfills an identity in the well-known ordinary way. Then, we have the following:

Proposition 3.

If the basic algebra $\mathcal{A}$ satisfies an identity $u\approx v$, then also the corresponding P-algebra fulfills this identity in the sense of (8).

Proof. 

Since the algebra $\mathcal{A}$ satisfies the identity, for all $x_1,...,x_n\in A$, $u(x_1,\dots ,x_n)=v(x_1,\dots ,x_n)$. Hence, $E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n))=\mu \left(u(a_1,\dots ,a_n)\right)$. By Corollary 2,

$$\bigcap _{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow \mu \left(u(a_1,\dots ,a_n)\right),$$

(9)

and hence

$$\bigcap _{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n)).$$

(10)

and P-algebra fulfills the identity $u\approx v$. □

The converse of the claim in Proposition 3 is not satisfied in general, although it is true in some special cases as follows.

Let $\phi (x_1,\dots ,x_n)$ be a term operation in the language of algebra $\mathcal{A}$. It is said to be idempotent on $\mathcal{A}$ if $\mathcal{A}$ satisfies the identity $\phi (x,x,\dots ,x)\approx x$.

Theorem 7.

A P-algebra $(\mathcal{A},E)$ satisfies an idempotent identity

$$\phi (x,x,\dots ,x)\approx x$$

if and only if this identity holds on the underlying algebra $\mathcal{A}$.

Proof. 

Suppose that a P-algebra $(\mathcal{A},E)$ satisfies an idempotent identity $\phi (x,x,\dots ,x)\approx x$ i.e., $\downarrow E(a,a)\subseteq \downarrow E\left(\phi \right(a,\dots ,a),a)$ for all $a\in A$. It follows that $E(a,a)⩽E\left(\phi \right(a,\dots ,a),a)$ (since $E(a,a)$ belongs to $\downarrow E(a,a)$ so it belongs also to $\downarrow E\left(\phi \right(a,\dots ,a),a)$). Now, by the separability property $\phi (a,\dots ,a)=a$ for all $a\in A$. The other implication is true by Proposition 3. □

Observe that by Theorem 4 $\left(ii\right)$, for every $p\in P$, ${\mu }_p$ is a subalgebra of the underlying algebra $\mathcal{A}$, and $E_p$ is a congruence relation on ${\mu }_p$. Therefore, for every $p\in P$, ${\mu }_p/E_p$ is a quotient structure over the subalgebra ${\mu }_p$.

Theorem 8.

A P-algebra $(\mathcal{A},E)$ satisfies an identity $u\approx v$ in the sense of (8), if and only if for every $p\in P$ the quotient subalgebra ${\mu }_p/E_p$ satisfies this identity in the classical way.

Proof. 

Suppose that P-algebra $(\mathcal{A},E)$ satisfies an identity $u(x_1,\cdots ,x_n)\approx v(x_1,\cdots ,x_n)$, where $x_1,...,x_n$ are all variables appearing in u and v. Then for any $a_1,\dots ,a_n\in A$

$$\bigcap _{i=1}^n\downarrow \mu \left(a_i\right)\subseteq \downarrow E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n)),$$

where $\mu \left(a_i\right)=E(a_i,a_i)$.

Let $p\in P$ and let $\left[a_1\right],...,\left[a_n\right]\in {\mu }_p/E_p$. We have that $a_1,...,a_n\in {\mu }_p$ and hence $\mu \left(a_i\right)⩾p$ for all $i\in \{1,...,n\}.$

Therefore

$$p\in \bigcap _{i=1}^n\downarrow \mu \left(a_i\right)$$

and hence

$$p\in \downarrow E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n)).$$

Consequently, $E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n))⩾p$ and $(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n))\in E_p$. Then finally

$${\left[u(a_1,\dots ,a_n)\right]}_{E_p}={\left[v(a_1,\dots ,a_n)\right]}_{E_p}.$$

(11)

So, the cut-quotient structures satisfy this identity in the classical way.

Conversely, suppose that for every $p\in P$, the quotient subalgebra ${\mu }_p/E_p$ of the given P-algebra $(\mathcal{A},E)$ satisfies this identity in the classical way, i.e., suppose that (11) holds for every $p\in P$. Then, for every p, we have $E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n))⩾p$ and therefore $p\in \downarrow E(u(a_1,\dots ,a_n),v(a_1,\dots ,a_n)).$ □

Analogously as for classical algebras, a P-algebra $(\mathcal{A},E)$satisfies a set of identities in the language of $\mathcal{A}$, if for each of this identities the Formula (8) holds.

Example 2.

Let $\mathcal{A}=\left(\right\{a,b,c\},·)$ be a three-element algebra, i.e., a three-element groupoid with a binary operation given in

Table 2. Operation ·.

·	a	b	c
a	a	b	c
b	a	b	c
c	c	c	c

.

Obviously, the operation in Table 2 is not commutative. Using the poset $(P,⩽)$, represented by the diagram in Figure 2, we define a P-valued equality E, as given in

Table 3. P-valued equality E.

E	a	b	c
a	q	s	t
b	s	p	0
c	t	0	r

.

Now, the P-algebra $(\mathcal{A},E)$ is a commutative P-groupoid, since the set-theoretic Formula (8), i.e.,

$$\downarrow \mu \left(x\right)\cap \downarrow \mu \left(y\right)\subseteq \downarrow E(x·y,y·x),$$

where $x,y\in \{a,b,c\}$ and $\mu \left(x\right)=E(x,x)$, is fulfilled.

E.g., since $\mu \left(a\right)=E(a,a)=q$ and similarly $\mu \left(c\right)=r$, the identity $x·y\approx y·x$ applied to a and c in the above formula, gives

$$\downarrow \mu \left(a\right)\cap \downarrow \mu \left(c\right)=\{q,s,t,v,0\}\cap \{r,s,t,v,0\}=\{s,t,v,0\}\subseteq \{r,s,t,v,0\}=\downarrow E(a·c,c·a).$$

The same holds for the other pairs of elements.

Equivalently, the P-algebra $(\mathcal{A},E)$ is a commutative P-groupoid by Theorem 8, since all the cut-quotient groupoids are commutative:

Recall that for (any) $w\in P$ the cut $E_w$ is a weak congruence on $A=\{a,b,c\}$, such that for $x,y\in A$, $(x,y)\in E_w$ if $E(x,y)⩾w$. This cut-relation is actually a congruence on the subalgebra ${\mu }_w$ of $\mathcal{A}$, obtained as the cut of the diagonal of E: $x\in {\mu }_w$ if $\mu \left(x\right)=E(x,x)⩾w.$ Following these definitions, we obtain the quotient structures ${\mu }_w/E_w,w\in P$, given as the collections of classes:

${\mu }_1/E_1={\mu }_u/E_u=\varnothing$; ${\mu }_p/E_p=\left\{\left\{b\right\}\right\}$; ${\mu }_q/E_q=\left\{\left\{a\right\}\right\}$; ${\mu }_r/E_r=\left\{\left\{c\right\}\right\}$; ${\mu }_s/E_s=\{\{a,b\},\left\{c\right\}\}$; ${\mu }_t/E_t={\mu }_v/E_v=\left\{\{a,c\}\right\}$; ${\mu }_0/E_0=\left\{\{a,b,c\}\right\}$.

Each of the above quotient groupoids consists of one or two elements - classes and they are commutative.

Essential algebraic properties of P-algebras are related to the quotient structures of their cut-subalgebras over the cuts of the P-valued equality E. Let $P1$ and $P2$ be posets and let $\mathcal{A}$ be an algebra. Suppose further that $(\mathcal{A},E1)$ is a $P1$-valued algebra and $(\mathcal{A},E2)$ a $P2$-valued algebra. We say that $(\mathcal{A},E1)$ and $(\mathcal{A},E2)$ are cut-equivalent if their collections of the mentioned quotient subalgebras over cuts of $E1$ and $E2$ coincide. This means that for every $p\in P1$ there is $q\in P2$ such that $\mu 1_p/E{1}_p=\mu 2_q/E{2}_q$ and vice versa.

Theorem 9.

Let $(\mathcal{A},E)$ be a P-algebra. Let also $P1$ be the collection of cuts of E ordered dually to the set inclusion:

$$P1=(\{\theta \in {\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)\mid \theta =E_p,\mathit{for}\mathit{some}p\in P\},\supseteq ).$$

Define $E1:A^2\to P1$:

$$E1(a,b):=\bigcap (\theta \in P1\mid (a,b)\in \theta )\mathit{for}\mathit{all}a,b\in A.$$

(12)

Then, $(\mathcal{A},E1)$ is a $P1$-algebra and P-algebra $(\mathcal{A},E)$ and $P1$-algebra $(\mathcal{A},E1)$ are cut-equivalent. In addition, every $\theta \in P1$ coincides with the corresponding cut, i.e., $E{1}_{\theta }=\theta$.

Proof. 

$P1$ is a set of some weak congruences of algebra $\mathcal{A}$ (congruences on subalgebras) by Theorem 4. It is a poset by the relation opposite of inclusion. Since $(\mathcal{A},E)$ is a P-algebra, the following property is satisfied by Proposition 2: for elements $a,b$, $E(a,a)\ne 0$ (if 0 exists in P),

$$ifa\ne bthen(a,b)\notin \bigcap \{\rho \in P1\mid (a,a)\in \rho \}.$$

(13)

In case there is no 0 in P or if there is $0\in P$ but there is no $a\in A$ such that $E(a,a)=0$, $(\mathcal{A},E1)$ is a $P1$-set by Theorem 2.

Suppose that there is $0\in P$ and $a\in A$ such that $E(a,a)=0$. Then, $E(a,b)=0$ for all $b\in A$. For such elements a, and all $b\in A$, $E1(a,b)=E_0,$ since there are no other cuts of E containing $(a,b)$. Similarly, we have that $E1(a,a)=E_0$, so $E1$ is separable.

Now, we prove that $E{1}_{\theta }=\theta$ for $\theta \in P1$. Let $(x,y)\in E{1}_{\theta }$. This is equivalent to $E1(x,y)⩾\theta$ by the definition of a cut. This is further equivalent to $E1(x,y)\subseteq \theta$ since the order in $P1$ is opposite to the inclusion. Further, we look at the definition of $E1$, and every $(x,y)$ belongs to $E1(x,y)$, so we have $(x,y)\in \theta$. To prove the opposite, suppose that $(x,y)\in \theta$. By the definition of $E1$, $E1(x,y)\subseteq \theta$ and $(x,y)\in E{1}_{\theta }$.

Finally, by Theorem 5 $(\mathcal{A},E1)$ is a $P1$-algebra since it has the same cuts as $(\mathcal{A},E)$. □

Theorem 9 shows that from the aspect of cut quotient structures; every P-algebra $(\mathcal{A},E)$ with a fixed basic algebra $\mathcal{A}$ and an arbitrary co-domain poset P can be identified in the weak congruence lattice ${\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$ of $\mathcal{A}$ by the suitable centralized system. Moreover, every centralized system of weak congruences of $\mathcal{A}$ satisfying condition (13) always determines a P-algebra over the corresponding quotients, which follows from Theorems 2, 8 and 9:

Theorem 10.

Let $\mathcal{A}$ be an algebra and $\mathcal{C}\subseteq {\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$ a centralized system of weak congruences of $\mathcal{A}$, satisfying

$$ifa\ne bthen(a,b)\notin \bigcap \{\rho \in \mathcal{F}\mid (a,a)\in \rho \}.$$

(14)

Let further $P=(\mathcal{C},\supseteq )$ and let

$$E(x,y)=\bigcap \{\rho \in \mathcal{C}\mid (x,y)\in \rho \}.$$

Then $(\mathcal{A},E)$ is a P-algebra, such that for every $p\in P$, $E_p=p$.

An identity in the language of $\mathcal{A}$ holds in $(\mathcal{A},E)$, if and only if each of the quotient algebras obtained by a weak congruence from $\mathcal{C}$ on the corresponding subalgebra of $\mathcal{A}$ satisfy this identity in the classical sense.

According to the above-defined cut-equivalence of P-algebras, we say that two cut-equivalent P-algebras over the same basic algebra $\mathcal{A}$ are equal up to the cut-equivalence. Consequently, different P-algebras over the same basic algebra are those that are not cut-equivalent.

Based on the definitions and properties of poset-valued structures elaborated in this section, we obtain the following practical conclusion: If an algebra $\mathcal{A}$ is given, then there are two main procedures to obtain a P-algebra $(\mathcal{A},E)$.

The first is to have an algebra $\mathcal{A}$ and also a poset $(P,⩽)$ and to construct a symmetric and transitive map $E:A^2\to P$, which is compatible with the operation in $\mathcal{A}$ in the sense of (4). If we want $(\mathcal{A},E)$ to also fulfill the set of identities $\mathcal{S}$, then for each of these identities, the Formula (8) is assumed to hold. This procedure is a straightforward application of the definition of a P-algebra, including the Formula (8) related to identities if necessary.

This procedure is applied in the construction of the P-valued commutative groupoid in Example 2.

In order to explain the second procedure, we recall that P-algebras are characterized by the centralized collections of weak congruences, which form a subposet in the corresponding lattice. These weak congruences determine the quotient structures over subalgebras. Clearly, all these subalgebras and quotient structures are of the same type as the starting basic algebra. Now, in the second procedure for getting a P-algebra starting with the classical algebra $\mathcal{A}$, the poset $(P,⩽)$ is not given in advance. Instead, a centralized system $\mathcal{C}$—a subposet of the lattice ${\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$ of weak congruences on $\mathcal{A}$, ordered dually to inclusion—is chosen to be the poset P. $\mathcal{C}$ consists of congruences on subalgebras of $\mathcal{A}$ and consequently, the corresponding quotient (sub)algebras are determined. This construction of a P-algebra is based on Theorem 10. Accordingly, if $\mathcal{A}$ satisfies some identities, then the quotient structures from $\mathcal{C}$ should also satisfy the corresponding identities.

The second procedure is explained in the sequel on the same three-element groupoid from Example 2.

Example 3.

Let $\mathcal{A}=\left(\right\{a,b,c\},·)$, $(P,⩽)$ and $(\mathcal{A},E)$ be as in Example 2.

The ∼-classes on the poset P, induced by the mapping E, are indicated on the diagram in Figure 3.

In Figure 4, there is the diagram of the weak congruence lattice ${\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$ of the above three-element algebra $\mathcal{A}$. The subposet consisting of the filled circles corresponds to the centralized system of weak congruences determined by the cuts of the P-valued equality E. When ordered dually to the set inclusion, this subposet is order-isomorphic to the poset $(P/\sim ,⩽)$ of ∼-classes, indicated on the poset $(P,⩽)$ in Figure 3 and represented by the diagram in Figure 5.

So, by the above-described procedures for constructing P-algebras, we should investigate the structure of the quotient subalgebras in the lattice ${\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$ of weak congruences of the starting basic algebra $\mathcal{A}$.

As already pointed out, if $\theta$ is a weak congruence on an algebra $\mathcal{A}$, i.e., if $\theta \in {\mathsf{Con}}_{\mathsf{w}}\left(\mathcal{A}\right)$, then $\theta$ can be considered as a congruence on the subalgebra $\mathcal{B}$ of $\mathcal{A}$, where $B=\{x\in A\mid (x,x)\in \theta \}$. Therefore, instead of analyzing the set of quotient structures of the form $B/\theta$, we use the collection of the corresponding weak congruences. They are relations $\theta$ which are in one-to-one correspondences to $B/\theta$.

The above analysis shows that for a given algebra $\mathcal{A}$, whichever poset P is chosen and the generalized equality E obtained as a map from $A^2$ to P, the obtained P-algebra is cut-equivalent to the one constructed by the suitable centralized system of weak congruences of $\mathcal{A}$. In this case, the poset $P/\sim$ is dually order isomorphic with the corresponding centralized system. Therefore, when dealing with P-algebras, we do not necessarily fix the poset P, but only indicate that the given basic algebra $\mathcal{A}$ is used for the construction of a poset-valued one $(\mathcal{A},E)$ by the construction via a centralized system of weak congruences.

4. Discussion and Conclusions

P-algebras, introduced here, are a generalization of classical algebras so that the ordinary equality in these algebras is replaced by the poset-valued function, where the poset P is chosen in advance. In order to check the validity of identities in the language of the starting (called basic) algebra, we use particular set-theoretic formulas. We have shown here that these new algebraic structures keep a close connection to classical algebras of the same type over the cut-quotient subalgebras. The canonical representation of these algebras that we define shows that information and properties obtained by using any poset as a co-domain of the generalized equality can be obtained by replacing the poset with a special subposet of the weak congruence lattice of the starting algebra.

As mentioned above, any class of P-algebras fulfilling known properties (e.g., identities) includes the classical algebras fulfilling the same properties as a subclass. Therefore, they are convenient for applications, particularly when the classical equality cannot be used (e.g., corrupted or missing data). Consequently, our further research in this field would be to investigate P-groups, P-moduls and P-vector-spaces and to apply them in solving linear equations and systems of these equations in the mentioned conditions. Another ample field of possible application of our newly introduced structure is analyzing relational data and also decision-making, where all the options are not linearly ordered.

As we already mentioned, this manuscript provides a generalization of some of the results on $\mathrm{\Omega }$-algebras, where $\mathrm{\Omega }$ is a complete lattice [10,11]. Recently, more techniques have been developed to address similar applications. Regarding the approximate solving of linear equations, in paper [16], approximate systems of linear equations are solved using the fuzzy and interval finite element methods. Particular methods of finding minimal solutions of generalized fuzzy relational equations are discussed in [17]. Formal concept analysis, as one of the most popular methods based on lattice theory used to extract knowledge from relational data [18], has recently been further developed to accommodate special applications [19,20]. Finally, the decision algorithms within the framework of rough sets have been developed in [21].