Subtraction Menger algebras

Abstract

We give an abstract characterization of algebras of partial functions from A ⁿ to A endowed with the operations of the Menger superposition and the set-theoretic difference of functions as subsets of A ⁿ⁺¹.

  1. Let A ⁿ be the n-th Cartesian power of a set A. Any partial mapping from A ⁿ into A is called a partial n-place function. The set of all such mappings is denoted by $\mathcal {F}(A^{n},A)$. On $\mathcal{F}(A^{n},A)$ we define the Menger superposition (composition) of n-place functions O:(f,g ₁,…,g _n )↦f[g ₁…g _n ] as follows:

$$ (\bar{a},c)\in f[g_1\ldots g_n]\longleftrightarrow(\exists\bar{b}) \bigl((\bar{a},b_1)\in g_1\wedge\cdots\wedge(\bar{a},b_n)\in g_n\wedge(\bar{b},c)\in f \bigr)$$

(1)

for all $\bar{a}\in A^{n}$, $\bar{b}=(b_{1},\ldots,b_{n})\in A^{n}$, c∈A.

Each subalgebra (Φ,O), where $\varPhi\subset\mathcal{F}(A^{n},A)$, of the algebra $(\mathcal{F}(A^{n},A),\mathrm{O})$ is a Menger algebra of rank n in the sense of [2–4, 8]. Menger algebras of partial n-place functions are partially ordered by the set-theoretic inclusion, i.e., such algebras can be considered as algebras of the form (Φ,O,⊂). The first abstract characterization of such algebras was given in [9]. Later, in [10, 11] there have been found abstract characterizations of Menger algebras of n-place functions closed with respect to the set-theoretic intersection and union of functions, i.e., Menger algebras of the form (Φ,O,∩), (Φ,O,∪) and (Φ,O,∩,∪).

As is well known, the set-theoretic inclusion ⊂ and the operations ∩, ∪ can be expressed via the set-theoretic difference (subtraction) in the following way:

where A,B,C are arbitrary sets such that A⊂C and B⊂C.

Thus it makes sense to examine sets of functions closed with respect to the subtraction of functions. Such sets of functions are called difference semigroups, while their abstract analogs are called subtraction semigroups. Some properties of subtraction semigroups can bee found in [1]. The investigation of difference semigroups was initiated by Schein [7].

Below we present a generalization of Schein’s results to the case of Menger algebras of n-place functions, i.e., to the case of algebras (Φ,O,∖,∅), where $\varPhi\subset\mathcal{F}(A^{n}, A)$, ∅∈Φ. Such algebras will be called difference Menger algebras.

2. A Menger algebra of rank n is a non-empty set G with one (n+1)-ary operation o(x,y ₁,…,y _n )=x[y ₁…y _n ] satisfying the identity:

$$ x [y_1 \ldots y_n] [z_1\ldots z_n] = x \bigl[y_1 [z_1\ldots z_n]\ldots y_n [z_1\ldots z_n]\bigr].$$

(2)

A Menger algebra of rank 1 is a semigroup. A Menger algebra (G,o) of rank n is called unitary if it contains selectors, i.e., elements e ₁,…,e _n ∈G such that x[e ₁…e _n ]=x and e _i [x ₁…x _n ]=x _i for all x,x ₁,…,x _n ∈G, i=1,…,n. One can prove (see [2, 3]) that every Menger algebra (G,o) of rank n can be isomorphically embedded into a unitary Menger algebra (G ^∗,o ^∗) of the same rank with selectors e ₁,…,e _n ∉G such that G∪{e ₁,…,e _n } is a generating set of (G ^∗,o ^∗).

Let (G,o) be a Menger algebra of rank n. Consider the alphabet G∪{[ , ],x}, where the symbols [ , ],x do not belong to G, and construct the set T _n (G) of polynomials over this alphabet by the following rules:

• (a) x∈T _n (G);

• (b) if i∈{1,…,n}, a,b ₁,…,b _i−1,b _i+1,…,b _n ∈G, t∈T _n (G), then a[b ₁…b _i−1 t  b _i+1,…b _n ]∈T _n (G);

• (c) T _n (G) contains those and only those polynomials which are constructed by (a) and (b).

A binary relation ρ⊂G×G, where (G,o) is a Menger algebra of rank n, is

• stable if for all x,y,x _i ,y _i ∈G, i=1,…,n

  $$(x,y),(x_1,y_1),\ldots,(x_n,y_n)\in\rho\longrightarrow \bigl(x[x_1\ldots x_n],y[y_1\ldots y_n]\bigr)\in\rho;$$

• l-regular, if for any x,y,z _i ∈G, i=1,…,n

  $$(x,y)\in\rho\longrightarrow\bigl(x[z_1\ldots z_n],y[z_1\ldots z_n]\bigr)\in\rho;$$

• v-regular, if for all x _i ,y _i ,z∈G, i=1,…,n

  $$(x_1,y_1),\ldots,(x_n,y_n)\in \rho\longrightarrow\bigl(z[x_1\ldots x_n],z[y_1\ldots y_n]\bigr)\in\rho;$$

• i-regular (1≤i≤n), if for all u,x,y∈G, $\bar{w}\in G^{n}$

  $$(x,y)\in\rho\longrightarrow\bigl(u[\bar{w}|_ix], u[\bar{w}|_iy]\bigr)\in\rho;$$

• weakly steady if for all x,y,z∈G, t ₁,t ₂∈T _n (G)

  $$(x,y),\bigl(z,t_1(x)\bigr),\bigl(z,t_2(y)\bigr)\in\rho \longrightarrow \bigl(z,t_2(x)\bigr)\in\rho,$$

where $\bar{w}=(w_{1},\ldots,w_{n})$ and $u[\bar{w}|_{i}\,x]=u[w_{1}\ldots w_{i-1}xw_{i+1}\ldots w_{n}]$. It is clear that a quasiorder¹ on a Menger algebra is v-regular if and only if it is i-regular for every i=1,…,n. A quasiorder is stable if and only if it is both v-regular and l-regular.

A subset H of a Menger algebra (G,o) is called

• stable if

  $$g,g_1,\ldots,g_n\in H\longrightarrow g[g_1\ldots g_n]\in H;$$

• an l-ideal, if for all x,h ₁,…,h _n ∈G

  $$(h_1,\ldots,h_n)\in G^n \setminus(G\setminus H)^n \longrightarrow x [h_1\ldots h_n]\in H;$$

• an i-ideal (1≤i≤n), if for all h,u∈G, $\bar{w}\in G^{n}$

  $$h\in H\longrightarrow u[\bar{w}|_i h]\in H.$$

Clearly, H is an l-ideal if and only if it is an i-ideal for every i=1,…,n.

FormalPara Definition 1

An algebra (G,−,0) of type (2,0) is called a subtraction algebra if it satisfies the following identities:

(3)

(4)

(5)

(6)

FormalPara Proposition 1

(Abbott [1])

Every subtraction algebra satisfies the identity

$$ 0=x-x.$$

(7)

FormalPara Proof

Below we give a short proof of this identity:

as required. □

From (7), by using (3), we obtain the following two identities:

$$ x-0 = x,\qquad0-x = 0.$$

(8)

Similarly, from (4), (5), (7) and (8) we can deduce the identities

(9)

(10)

Thus, subtraction algebras are implicative BCK-algebras (cf. [5, 6]).

FormalPara Definition 2

An algebra (G,o,−,0) of type (n+1,2,0) is called a subtraction Menger algebra of rank n, if (G,o) is a Menger algebra of rank n, (G,−,0) is a subtraction algebra and the conditions

(11)

(12)

(13)

hold for all x,y,z,u,z ₁,…,z _n ∈G, $\bar{w}\in G^{n}$, i=1,…,n and t ₁,t ₂∈T _n (G).

By putting n=1 in the above definition we obtain the notion of a weak subtraction semigroup ² studied by Schein (cf. [7]). Such semigroups are isomorphic to some subtraction semigroups of the form (Φ,∘,∖).

3. Now we can present the first result of our paper.

FormalPara Theorem 1

Each difference Menger algebra of n-place functions is a subtraction Menger algebra of rank n.

FormalPara Proof

Let (Φ,O,∖,∅) be a difference Menger algebra of n-place functions defined on A. Since, as it is proved in [2], the superposition O satisfies (2), the algebra (Φ,O) is a Menger algebra of rank n. From the results proved in [1] it follows that the operation ∖ satisfies (3), (4) and (5). Hence (Φ,∖,∅) is a subtraction algebra. Thus, (Φ,O,∖,∅) will be a subtraction Menger algebra if (11), (12) and (13) will be satisfied.

To verify (11) observe that for each $(\bar{a},c)\in(f\setminus g)[h_{1}\ldots h_{n}]$, where f,g,h ₁,…,h _n ∈Φ, $\bar{a}\in A^{n}$, c∈A there exists $\bar{b}=(b_{1},\ldots,b_{n})\in A^{n}$ such that $(\bar{b},c)\in f \setminus g$ and $(\bar{a},b_{i})\in h_{i}$ for each i=1,…,n. Consequently, $(\bar{b},c)\in f$ and $(\bar{b},c)\notin g$. Thus, $(\bar{a},c)\in f[h_{1}\ldots h_{n}]$. If $(\bar{a},c)\in g[h_{1}\ldots h_{n}]$, then there exists $\bar {d}=(d_{1},\ldots, d_{n})\in A^{n}$ such that $(\bar{d},c)\in g$ and $(\bar {a},d_{i})\in h_{i}$ for every i=1,…,n. Since h ₁,…,h _n are functions, we obtain b _i =d _i for all i=1,…,n. Thus $\bar{b}=\bar{d}$. Therefore $(\bar{b},c)\in g$, which is impossible. Hence $(\bar {a},c)\notin g[h_{1}\ldots h_{n}]$. This means that $(\bar{a},c)\in f[h_{1}\ldots h_{n}]\setminus g[h_{1}\ldots h_{n}]$. So, the following implication

$$(\bar{a},c)\in(f \setminus g)[h_1\ldots h_n]\longrightarrow (\bar{a},c)\in f[h_1\ldots h_n]\setminus g[h_1\ldots h_n]$$

is valid for any $\bar{a}\in A^{n}$, c∈A, i.e., (f∖g)[h ₁…h _n ]⊂f[h ₁…h _n ]∖g[h ₁…h _n ].

Conversely, let $(\bar{a},c)\in f[h_{1}\ldots h_{n}]\setminus g[h_{1}\ldots h_{n}]$. Then $(\bar{a},c)\in f[h_{1}\ldots h_{n}]$ and $(\bar{a},c)\notin g [h_{1}\ldots h_{n}]$. Thus, there exists $\bar{b}=(b_{1},\ldots,b_{n})\in A^{n}$ such that $(\bar{b},c)\in f$, $(\bar{b},c)\notin g$ and $(\bar{a},b_{i})\in h_{i}$ for each i=1,…,n. Hence, $(\bar{b},c)\in f \setminus g$ and $(\bar{a},c)\in(f \setminus g)[h_{1}\ldots h_{n}]$. So,

$$(\bar{a},c)\in f[h_1\ldots h_n]\setminus g[h_1\ldots h_n]\longrightarrow(\bar{a},c)\in(f \setminus g)[h_1\ldots h_n]$$

for any $\bar{a}\in A^{n}$, c∈A, i.e., f[h ₁…h _n ]∖g[h ₁…h _n ]⊂(f∖g)[h ₁…h _n ]. Thus,

$$(f \setminus g)[h_1\ldots h_n]=f[h_1\ldots h_n]\setminus g[h_1\ldots h_n],$$

which proves (11).

Now, let $(\bar{a},c)\in u[\bar{\omega}|_{i}(f \setminus(f \setminus g))] =u[\bar{\omega}|_{i}(f\cap g)]$, where f,g,u∈Φ, $\bar{\omega}\in\varPhi^{n}$, $\bar{a}\in A^{n}$, c∈A. Then there exists $\bar{b}=(b_{1},\ldots,b_{n})\in A^{n}$ such that $(\bar{a},b_{i})\in f\cap g$, $(\bar{a},b_{j})\in\omega_{j}$, j∈{1,…,n}∖{i} and $(\bar{b},c)\in u$. Since $(\bar{a},b_{i})\in f\cap g$ implies $(\bar{a},b_{i})\notin f \setminus g$, we have $(\bar{a},c)\in u[\bar {\omega}|_{i}f]$ and $(\bar{a},c)\notin u[\bar{\omega}|_{i}(f \setminus g)]$. Therefore $(\bar{a},c)\in u[\bar{\omega}|_{i}f]\setminus u[\bar {\omega}|_{i}(f \setminus g)]$. Thus, we have shown that for any $\bar{a}\in A^{n}$, c∈A holds the implication

$$(\bar{a},c)\in u\bigl[\bar{\omega}|_i\bigl(f \setminus(f \setminus g)\bigr)\bigr]\longrightarrow(\bar{a},c)\in u[\bar{\omega}|_if]\setminus u\bigl[\bar{\omega}|_i(f \setminus g)\bigr],$$

which is equivalent to the inclusion $u[\bar{\omega}|_{i}(f \setminus(f \setminus g))]\subset u[\bar{\omega}|_{i}f]\setminus u[\bar{\omega}|_{i}(f \setminus g)]$.

Conversely, let $(\bar{a},c)\in u[\bar{\omega}|_{i}f]\setminus u[\bar {\omega}|_{i}(f \setminus g)]$. Then $(\bar{a},c)\in u[\bar{\omega}|_{i}f]$ and $(\bar{a},c)\notin u[\bar{\omega}|_{i}(f \setminus g)]$. The first of these two conditions means that there exists $\bar{b}=(b_{1},\ldots,b_{n})\in A^{n}$ such that $(\bar{a},b_{i})\in f$, $(\bar {a},b_{j})\in\omega_{j}$ for each j∈{1,…,n}∖{i} and $(\bar{b},c)\in u$. It is easy to see that the second condition $(\bar{a},c)\notin u[\bar{\omega}|_{i}(f \setminus g)]$ is equivalent to the implication

$$ (\forall\bar{d}) \Biggl((\bar{a},d_i)\in f\wedge \bigwedge_{j=1,j\neq i}^{n}(\bar{a},d_j)\in\omega_j \wedge(\bar{d},c)\in u\longrightarrow(\bar{a},d_i)\in g \Biggr),$$

(14)

where $\bar{d}=(d_{1},\ldots,d_{n})\in A^{n}$. From this implication for $\bar{d}=\bar{b}$, we obtain

$$(\bar{a},b_i)\in f\wedge\bigwedge_{j=1,j\neq i}^n(\bar{a},b_j)\in\omega_j\wedge(\bar{b},c)\in u\longrightarrow (\bar{a},b_i)\in g,$$

which gives $(\bar{a},b_{i})\in g$. Therefore $(\bar{a},b_{i})\in f\cap g=f \setminus(f \setminus g)$. This means that $(\bar{a},c)\in u[\bar{\omega}|_{i}(f \setminus(f \setminus g))]$. So, the implication

$$(\bar{a},c)\in u[\bar{\omega}|_if]\setminus u\bigl[\bar{\omega}|_i(f \setminus g)\bigr]\longrightarrow(\bar{a},c)\in u\bigl[\bar{\omega}|_i\bigl(f \setminus(f \setminus g)\bigr)\bigr]$$

is valid for all $\bar{a}\in A^{n}$, c∈A. Hence $u[\bar{\omega}|_{i}f]\setminus u[\bar{\omega}|_{i}(f \setminus g)]\subset u[\bar{\omega}|_{i}(f \setminus(f \setminus g))]$. Thus

$$u\bigl[\bar{\omega}|_i\bigl(f \setminus(f \setminus g)\bigr)\bigr]=u[\bar{\omega}|_if]\setminus u\bigl[\bar{\omega}|_i(f\setminus g)\bigr].$$

This proves (12).

To prove (13) suppose that for some f,g,h∈Φ and t ₁,t ₂∈T _n (Φ) we have f∖g=∅, h∖t ₁(f)=∅ and h∖t ₂(g)=∅. Then f⊂g, h⊂t ₁(f) and h⊂t ₂(g). Hence $f=g\circ\Delta_{{\operatorname {pr_{1}}}f}$ and $\operatorname {pr_{1}}h\subset \operatorname {pr_{1}}f$, where $\operatorname {pr_{1}}f$ denotes the domain of f and $\Delta_{{\operatorname {pr_{1}}}f}$ is the identity binary relation on $\operatorname {pr_{1}}f$.

From the inclusion h⊂t ₂(g) we obtain

$$h=h\circ\Delta_{{\operatorname {pr_1}}f}\subset t_2(g)\circ \Delta_{{\operatorname {pr_1}}f}=t_2(g\circ\Delta_{{\operatorname {pr_1}}f})=t_2(f),$$

which means that (13) is also satisfied. This completes the proof that (Φ,O,∖,∅) is a subtraction Menger algebra of rank n. □

To prove the converse statement, we should first consider a number of properties of subtraction Menger algebras of rank n, introduce some definitions and prove a few auxiliary propositions.

4. Let (G,o,−,0) be a subtraction Menger algebra of rank n.

FormalPara Proposition 2

In every subtraction Menger algebra of rank n we have

$$0[x_1\ldots x_n]=0, \qquad x[x_1\ldots x_{i-1}0\,x_{i+1}\ldots x_n]=0$$

for all x,x ₁,…,x _n ∈G, i=1,…,n.

FormalPara Proof

Indeed, using (7) and (11) we obtain

$$0[x_1\ldots x_n]=(0-0)[x_1\ldots x_n]=0[x_1\ldots x_n]-0[x_1\ldots x_n]=0.$$

Similarly, applying (12) and (7) we get

$$u[\bar{w}|_i\,0]=u\bigl[\bar{w}|_i\bigl(0-(0-0)\bigr)\bigr]=u[\bar{w}|_i\,0]-u\bigl[\bar {w}|_i(0-0)\bigr]=u[\bar{w}|_i\,0]- u[\bar{w}|_i\,0]=0,$$

which was to show. □

Let ω be a binary relation defined on (G,o,−,0) in the following way:

$$\omega=\bigl\{(x, y)\in G\times G \mid x-y = 0\bigr\}.$$

Using (7), (8) and (9) it is easy to see that this is an order, i.e., a reflexive, transitive and antisymmetric relation. In connection with this fact we will sometimes write x≤y instead of (x,y)∈ω. Using this notation it is not difficult to verify that

(15)

(16)

(17)

(18)

(19)

holds for all x,y,z,u,v∈G.

Moreover, in a subtraction algebra the following two identities

(20)

(21)

are valid (cf. [1, 5, 6]).

FormalPara Proposition 3

The relation ω on the algebra (G,o,−,0) is stable and weakly steady.

FormalPara Proof

Let x≤y for some x,y∈G. Then x−y=0 and

$$(x-y)[z_1\ldots z_n]=0[z_1\ldots z_n]=(0-0)[z_1\ldots z_n]=0[z_1\ldots z_n]-0[z_1\ldots z_n]=0$$

for all z ₁,…,z _n ∈G. This, by (11), implies

$$x[z_1\ldots z_n]-y[z_1\ldots z_n]=0,$$

i.e., x[z ₁…z _n ]≤y[z ₁…z _n ]. Thus, ω is l-regular.

Moreover, from x≤y, using (8), we obtain x−(x−y)=x, which together with (4), gives y−(y−x)=x. Consequently, for any u∈G, $\bar{w}\in G^{n}$ we have $u[\bar {w}|_{i}(y-(y-x))]=u[\bar{w}|_{i}\,x]$. This and (11) give $u[\bar{w}|_{i}\,y]-u[\bar{w}|_{i}(y-x)]=u[\bar{w}|_{i}\,x]$. Hence, according to (15), we obtain $u[\bar{w}|_{i}\,x]\leq u[\bar{w}|_{i}\,y]$. Thus, ω is i-regular for every i=1,…,n. Since ω is a quasiorder, this means that ω is v-regular. But ω also is l-regular, hence it is stable.

It is clear that ω is weakly steady if and only if it satisfies (13).³ □

FormalPara Proposition 4

The axiom (12) is equivalent to each of the following conditions:

(22)

(23)

(24)

for all x,y,u∈G, $\bar{w}\in G^{n}$, i=1,…,n, t∈T _n (G).

FormalPara Proof

(12)→(22). Suppose that the condition (12) is satisfied and x≤y for some x,y∈G. Then, according to (16), we have x−(x−y)=x. Hence, by (4), we obtain y−(y−x)=x. Thus, y−x=y−(y−(y−x)), which, in view of (12), gives $u[\bar{w}|_{i}\,(y-x)]=u[\bar{w}|_{i}\,(y-(y-(y-x)))]=u[\bar{w}|_{i}y]-u[\bar {w}|_{i}(y-(y-x))]=u[\bar{w}|_{i}y]-u[\bar{w}|_{i}x]$. This means that (12) implies (22).

(22)→(23). From (22) it follows that for x≤y and all polynomials t∈T _n (G) of the form $t(x)=u[\bar{w}|_{i}x]$ the condition (23) is satisfied. To prove that (23) is satisfied by an arbitrary polynomial from T _n (G) suppose that it is satisfied by some t′∈T _n (G). Since the relation ω is stable on the algebra (G,o,−,0), from x≤y it follows t′(x)≤t′(y), which in view of (22), implies

$$u\bigl[\bar{w}|_i\,\bigl(t'(y)-t'(x)\bigr)\bigr]=u\bigl[\bar{w}|_i\,t'(y)\bigr]-u\bigl[\bar{w}|_i\,t'(x)\bigr].$$

But according to the assumption on t′ for x≤y we have t′(y)−t′(x)=t′(y−x), so the above equation can be written as

$$u\bigl[\bar{w}|_i\,t'(y-x)\bigr]=u\bigl[\bar{w}|_i\,t'(y)\bigr]-u\bigl[\bar{w}|_i\,t'(x)\bigr].$$

Thus, (23) is satisfied by polynomials of the form $t(x)=u[\bar{w}|_{i}t'(x)]$.

From the construction of T _n (G) it follows that (23) is satisfied by all polynomials t∈T _n (G). Therefore (22) implies (23).

(23)→(24). Since, by (15), x−y≤x holds for all x,y∈G, from (23) it follows t(x−(x−y))=t(x)−t(x−y) for any polynomial t∈T _n (G). Thus, (23) implies (24).

(24)→(12). By putting $t(x)=u[\bar{w}|_{i}\,x]$ we obtain (12). □

On a subtraction Menger algebra (G,o,−,0) of rank n we can define a binary operation ⋏ by putting:

$$ x\curlywedge y\stackrel{\mathit{def}}{=}x-(x-y).$$

(25)

By using this operation the conditions (11), (16), (24) can be written in a more useful form:

(26)

(27)

(28)

where x,y,u∈G, $\bar{w}\in G^{n}$, i=1,…,n, t∈T _n (G). Moreover, from (11) and (25), we can deduce the identity:

$$ (x\curlywedge y)[z_1\ldots z_n]=x[z_1\ldots z_n]\curlywedge y[z_1\ldots z_n].$$

(29)

The algebra (G,⋏) is a lower semilattice. Directly from the conditions (3)–(10) we obtain (cf. [1]) the following properties:

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

for all x,y,z∈G.

FormalPara Proposition 5

In a subtraction Menger algebra (G,o,−,0) of rank n the following conditions

(39)

(40)

are valid for each t∈T _n (G) and x,y∈G.

FormalPara Proof

From (35) we obtain t(x−y)=t(x−(x⋏y)) for every t∈T _n (G). (25) and (15) imply x⋏y≤x, which together with (23) gives t(x−(x⋏y))=t(x)−t(x⋏y). Hence, t(x−y)=t(x)−t(x⋏y). This proves (39).

Since x⋏y≤y, the stability of ω implies t(x⋏y)≤t(y) for every t∈T _n (G). From this, by applying (15) and (18), we obtain t(x)−t(y)≤t(x)−t(x⋏y)=t(x−y), which proves (40). □

By [0,a] we denote the initial segment of the algebra (G,−,0), i.e., the set of all x∈G such that 0≤x≤a. According to [7], on any [0,a] we can define a binary operation ⋎ by putting:

$$ x\curlyvee y\stackrel{\mathit{def}}{=}a-\bigl((a-x)\curlywedge(a-y)\bigr)$$

(41)

for all x,y∈[0,a]. It is not difficult to see that this operation is idempotent and commutative, and 0 is its neutral element, i.e., x⋎x=x, x⋎y=y⋎x, x⋎0=x for all x,y∈[0,a].

FormalPara Proposition 6

For any x,y∈[0,b]⊂[0,a], where a,b∈G, we have

$$ b-\bigl((b-x)\curlywedge(b-y)\bigr)=a-\bigl((a-x)\curlywedge(a-y)\bigr).$$

(42)

FormalPara Proof

Note first that b=b⋏a because b≤a. Moreover, from x≤b and y≤b, according to (18), we obtain a−b≤a−x and a−b≤a−y. This together with (30) gives a−b≤(a−x)⋏(a−y). Thus, (a−b)−((a−x)⋏(a−y))=0.

By (15) we have b−((a−x)⋏(a−y))≤b, which implies

$$ b\curlywedge\bigl(b-\bigl((a-x)\curlywedge(a-y)\bigr)\bigr)=b-\bigl((a-x)\curlywedge(a-y)\bigr).$$

(43)

Obviously b=b⋏b=b⋏a, x=b⋏x, y=b⋏y. Therefore:⁴

which completes the proof. □

FormalPara Corollary 1

The condition (42) is valid for all x,y∈[0,a]∩[0,b].

FormalPara Proof

Since [0,a]∩[0,b]=[0,a⋏b]⊂[0,a]∪[0,b], by Proposition 6, for all x,y∈[0,a]∩[0,b] we have:

This implies (42). □

From the above corollary it follows that the value of x⋎y, if it exists, does not depend on the choice of the interval [0,a] containing the elements x and y. In [1] it is proved that for x,y,z∈[0,a] we have:

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

From (44) it follows x≤x⋎y.

FormalPara Proposition 7

If for some x,y∈G there exists x⋎y, then for all u∈G, $\bar{z},\bar{w}\in G^{n}$, i=1,…,n there are also elements $x[\bar{z}]\curlyvee y[\bar{z}]$ and $u[\bar{w}|_{i}\,x]\curlyvee u[\bar{w}|_{i}\,y]$, and the following identities are satisfied:

(54)

(55)

FormalPara Proof

Suppose that the element x⋎y exists. Then x≤a and y≤a for some a∈G, which, by the l-regularity of the relation ω, implies $x[\bar{z}]\leq a[\bar{z}]$ and $y[\bar{z}]\leq a[\bar{z}]$ for any $\bar{z}\in G^{n}$. This means that $x[\bar{z}]\curlyvee y[\bar{z}]$ exists and

This proves (54).

Further, from x≤a, y≤a and the i-regularity of ω we obtain $u[\bar{w}|_{i}\,x]\leq u[\bar{w}|_{i}\,a]$ and $u[\bar{w}|_{i}\,y]\leq u[\bar{w}|_{i}\,a]$. Hence, the element $u[\bar {w}|_{i}\,x]\curlyvee u[\bar{w}|_{i}\,y]$ exists. Since x≤x⋎y and y≤x⋎y, we also have $u[\bar {w}|_{i}\,x]\leq u[\bar{w}|_{i}(x\curlyvee y)]$ and $u[\bar{w}|_{i}\,y]\leq u[\bar{w}|_{i}(x\curlyvee y)]$, which, according to (50), gives

$$ u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y]\leq u\bigl[\bar{w}|_i(x\curlyvee y)\bigr].$$

(56)

On the other side, the existence of $u[\bar{w}|_{i}\,x]\curlyvee u[\bar {w}|_{i}\,y]$ implies

$$u[\bar{w}|_i\,x]\leq u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y] \quad \mbox{and} \quad u[\bar{w}|_i\,y]\leq u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y].$$

Moreover,

$$u\bigl[\bar{w}|_i(x\curlyvee y)\bigr]-u\bigl[\bar{w}|_i(y-x)\bigr]\stackrel{\mbox{\scriptsize(40)}}{\leq} u\bigl[\bar{w}|_i\bigl((x\curlyvee y)-(y-x)\bigr)\bigr] \stackrel{\mbox{\scriptsize(52)}}{=}u[\bar{w}|_i\,x].$$

Consequently,

$$ u\bigl[\bar{w}|_i(x\curlyvee y)\bigr]-u\bigl[\bar{w}|_i(y-x)\bigr]\leq u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y].$$

(57)

But y−x≤y, so, $u[\bar{w}|_{i}(y-x)]\leq u[\bar{w}|_{i}\,y]$ and

$$u\bigl[\bar{w}|_i(y-x)\bigr]\leq u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y].$$

This and (57) guarantee the existence of the element

$$\bigl(u\bigl[\bar{w}|_i(x\curlyvee y)\bigr]-u\bigl[\bar{w}|_i(y-x)\bigr]\bigr)\curlyvee u\bigl[\bar{w}|_i(y-x)\bigr]$$

such that

$$ \bigl(u\bigl[\bar{w}|_i(x\curlyvee y)\bigr]-u\bigl[\bar{w}|_i(y-x)\bigr]\bigr)\curlyvee u\bigl[\bar{w}|_i(y-x)\bigr]\leq u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y].$$

(58)

Since $u[\bar{w}|_{i}(y-x)]\leq u[\bar{w}|_{i}\,y]\leq u[\bar{w}|_{i}(x\curlyvee y)]$, the last inequality and (51) imply

$$\bigl(u\bigl[\bar{w}|_i(x\curlyvee y)\bigr]-u\bigl[\bar{w}|_i(y-x)\bigr]\bigr)\curlyvee u\bigl[\bar{w}|_i(y-x)\bigr]=u\bigl[\bar{w}|_i(x\curlyvee y)\bigr],$$

which together with (58) gives

$$u\bigl[\bar{w}|_i(x\curlyvee y)\bigr]\leq u[\bar{w}|_i\,x]\curlyvee u[\bar{w}|_i\,y].$$

Comparing this inequality with (56) we obtain (55). □

FormalPara Corollary 2

If for some x,y∈G an element x⋎y exists, then for any polynomial t∈T _n (G) an element t(x)⋎t(y) also exists and t(x⋎y)=t(x)⋎t(y).

FormalPara Proposition 8

For all x,y∈G and all polynomials t ₁,t ₂∈T _n (G) we have:

$$t_1(x\curlywedge y)\curlywedge t_2(x-y)=0.$$

FormalPara Proof

Let t ₁(x⋏y)⋏t ₂(x−y)=h. Obviously h≤t ₁(x⋏y) and h≤t ₂(x−y). Since t ₂(x−y)≤t ₂(x), we have h≤t ₂(x). Thus, x⋏y≤x, h≤t ₁(x⋏y) and h≤t ₂(x). This, in view of Proposition 3 and (13), gives h≤t ₂(x⋏y). Consequently,

$$ h\leq t_2(x-y)\curlywedge t_2(x\curlywedge y).$$

(59)

Further,

Therefore,

which together with (59) implies h≤0. Hence h=0. This completes the proof. □

FormalPara Proposition 9

For all x,y,z,g∈G and all polynomials t ₁,t ₂∈T _n (G) the following conditions are valid:

(60)

(61)

(62)

FormalPara Proof

To prove (60) observe first that for z=t ₁(x⋏y)⋏t ₂(y) we have z≤t ₁(x⋏y) and z≤t ₂(y). Since the relation ω is weakly steady and x⋏y≤y, from the above we conclude z≤t ₂(x⋏y), i.e., t ₁(x⋏y)⋏t ₂(y)≤t ₂(x⋏y). This, by (31), implies t ₁(x⋏y)⋏t ₂(y)≤t ₁(x⋏y)⋏t ₂(x⋏y).

On the other side, the stability of ω and x⋏y≤y imply t ₂(x⋏y)≤t ₂(y) for every t ₂∈T _n (G). Hence, t ₁(x⋏y)⋏t ₂(x⋏y)≤t ₁(x⋏y)⋏t ₂(y) by (31). This completes the proof of (60).

Further: $t_{1}(x\curlywedge y\curlywedge z)\curlywedge t_{2}(y)=t_{1}((x\curlywedge z)\curlywedge y)\curlywedge t_{2}(y)\stackrel{\mbox{\mbox{\scriptsize(60)}}}{=}t_{1}((x\curlywedge z)\curlywedge y)\curlywedge t_{2}((x\curlywedge z)\curlywedge y)\leq t_{1}(x\curlywedge y)\curlywedge t_{2}(y\curlywedge z)$ proves (61).

Finally, let g≤t ₁(x⋏y) and g≤t ₂(y⋏z). Then

This proves (62) and completes the proof of our proposition. □

FormalPara Corollary 3

For all x,y,z∈G and all polynomials t ₁,t ₂∈T _n (G) we have:

$$ t_1(x\curlywedge y\curlywedge z)\curlywedge t_2(y)= t_1(x\curlywedge y)\curlywedge t_2(y\curlywedge z).$$

(63)

FormalPara Proof

We have t ₁(x⋏y)⋏t ₂(y⋏z)≤t ₁(x⋏y) and t ₁(x⋏y)⋏t ₂(y⋏z)≤t ₂(y⋏z), so by (62) we obtain t ₁(x⋏y)⋏t ₂(y⋏z)≤t ₁(x⋏y⋏z). Considering now that t ₁(x⋏y)⋏t ₂(y⋏z)≤t ₂(y⋏z)≤t ₂(y), by (30), we get t ₁(x⋏y)⋏t ₂(y⋏z)≤t ₁(x⋏y⋏z)⋏t ₂(y). Taking now into account the condition (61) we obtain (63). □

5. Let (G,o,−,0) be a subtraction Menger algebra of rank n.

FormalPara Definition 3

By a determining pair of a subtraction Menger algebra (G,o,−,0) of rank n we mean an ordered pair (ε ^∗,W), where ε is a v-regular equivalence relation defined on (G,o), ε ^∗=ε∪{(e ₁,e ₁),…,(e _n ,e _n )}, e ₁,…,e _n are the selectors of the unitary extension (G ^∗,o ^∗) of (G,o) and W is the empty set or an l-ideal of (G,o) which is an ε-class.

FormalPara Definition 4

A non-empty subset F of a subtraction Menger algebra (G,o,−,0) of rank n is called a filter if:

• (1) 0∉F;

• (2) x∈F∧x≤y⟶y∈F;

• (3) x∈F∧y∈F⟶x⋏y∈F

for all x,y∈G.

If a,b∈G and a≰b, then [ a)={x∈G | a≤x} is a filter with a∈[ a) and b∉[ a). By Zorn’s Lemma the collection of filters which contain an element a, but do not contain an element b, has a maximal element which is denoted by F _a,b . Using this filter we define the following three sets:

FormalPara Proposition 10

For any a,b∈G, the pair $(\varepsilon^{*}_{a,b},W_{a,b})$ is the determining pair of the algebra (G,o,−,0).

FormalPara Proof

First we show that ε _a,b is an equivalence relation on G. It is clear that this relation is reflexive and symmetric. To prove its transitivity let (x,y),(y,z)∈ε _a,b . We have four possibilities:

• (a) x⋏y∉W _a,b ∧ y⋏z∉W _a,b ,

• (b) x⋏y∉W _a,b ∧ y,z∈W _a,b ,

• (c) x,y∈W _a,b ∧ y⋏z∉W _a,b ,

• (d) x,y∈W _a,b ∧ y,z∈W _a,b .

In the case (a) we have t ₁(x⋏y),t ₂(y⋏z)∈F _a,b for some t ₁,t ₂∈T _n (G). Since F _a,b is a filter, then, obviously, t ₁(x⋏y)⋏t ₂(y⋏z)∈F _a,b . This, according to (63), implies t ₁(x⋏y⋏z)⋏t ₂(y)∈F _a,b . But t ₁(x⋏y⋏z)⋏t ₂(y)≤t ₁(x⋏z), hence also t ₁(x⋏z)∈F _a,b , i.e., x⋏z∉W _a,b . Thus, (x,z)∈ε _a,b .

In the case (b) from x⋏y∉W _a,b it follows t(x⋏y)∈F _a,b for some polynomial t∈T _n (G). But x⋏y≤y, and consequently t(x⋏y)≤t(y). Thus t(y)∈F _a,b , i.e., y∉W _a,b , which is a contradiction. Hence the case (b) is impossible. Analogously we can show that also the case (c) is impossible. The case (d) is obvious, because in this case x,z∈W _a,b which means that (x,z)∈ε _a,b . This completes the proof that ε _a,b is transitive.

Moreover, if x∈W _a,b , then t(x)∉F _a,b for every t∈T _n (G). In particular, for all $t(x)=t'(u[\bar{w}|_{i}\,x])\in T_{n}(G)$ we have $t'(u[\bar{w}|_{i}\,x])\notin F_{a,b}$. Thus, $u[\bar {w}|_{i}\,x]\in W_{a,b}$ for every i=1,…,n. Hence, W _a,b is an i-ideal of (G,o), and consequently, an l-ideal. It is clear that W _a,b is an ε _a,b -class.

Next, we prove that the relation ε _a,b is v-regular. Let x≡y(ε _a,b ). Then x⋏y∉W _a,b or x,y∈W _a,b . In the case x,y∈W _a,b we obtain $u[\bar{w}|_{i}\,x],u[\bar{w}|_{i}\,y]\in W_{a,b}$ because W _a,b is an l-ideal of (G,o). Thus, $u[\bar{w}|_{i}\,x]\equiv u[\bar{w}|_{i}\,y](\varepsilon_{a,b})$. In the case x⋏y∉W _a,b elements $u[\bar{w}|_{i}\,x]$, $u[\bar{w}|_{i}\,y]$ belong or not belong to W _a,b simultaneously. Indeed, if $u[\bar{w}|_{i}\,x]$, $u[\bar{w}|_{i}\,y]\in W_{a,b}$, then obviously $u[\bar{w}|_{i}\,x]\equiv u[\bar{w}|_{i}\,y](\varepsilon_{a,b})$. Now, if $u[\bar{w}|_{i}\,x]\notin W_{a,b}$, then $t(u[\bar{w}|_{i}\,x])\in F_{a,b}$ for some t∈T _n (G). Since x⋏y∉W _a,b , then also t ₁(x⋏y)∈F _a,b for some t ₁∈T _n (G). Thus $t_{1}(x\curlywedge y)\curlywedge t(u[\bar{w}|_{i}\,x])\in F_{a,b}$, which, by (60), implies $t_{1}(x\curlywedge y)\curlywedge t(u[\bar{w}|_{i}(x\curlywedge y)])\in F_{a,b}$. But $t_{1}(x\curlywedge y)\curlywedge t(u[\bar{w}|_{i}(x\curlywedge y)]) \leq t(u[\bar{w}|_{i}\,y])$, hence $t(u[\bar{w}|_{i}\,y])\in F_{a,b}$, i.e., $u[\bar{w}|_{i}\,y]\notin W_{a,b}$. So, we have shown that x⋏y∉W _a,b and $u[\bar{w}|_{i}\,x]\notin W_{a,b}$ imply $u[\bar{w}|_{i}\,y]\notin W_{a,b}$. Similarly we can show that x⋏y∉W _a,b and $u[\bar{w}|_{i}\,y]\notin W_{a,b}$ imply $u[\bar{w}|_{i}\,x]\notin W_{a,b}$. Therefore, we have proved that in the case x⋏y∉W _a,b elements $u[\bar{w}|_{i}\,x]$, $u[\bar{w}|_{i}\,y]$ belong or not belong to W _a,b simultaneously.

So, if for x⋏y∉W _a,b we have $u[\bar{w}|_{i}\,x],u[\bar{w}|_{i}\,y]\in W_{a,b}$, then clearly $u[\bar{w}|_{i}\,x]\equiv u[\bar{w}|_{i}\,y](\varepsilon_{a,b})$. Therefore assume that $u[\bar {w}|_{i}\,x]\notin W_{a,b}$ (hence $u[\bar{w}|_{i}\,y]\notin W_{a,b}$). Thus, x⋏y∉W _a,b , $u[\bar{w}|_{i}\,x]\notin W_{a,b}$, i.e., t(x⋏y)∈F _a,b , $t_{1}(u[\bar{w}|_{i}\,x])\in F_{a,b}$ for some t,t ₁∈T _n (G). Hence, $t(y\curlywedge x\curlywedge y)\curlywedge t_{1}(u[\bar{w}|_{i}\,x])\in F_{a,b}$. From this, according to (63), we obtain $t(y\curlywedge x)\curlywedge t_{1}(u[\bar{w}|_{i}(x\curlywedge y)])\in F_{a,b}$. This implies $t_{1}(u[\bar{w}|_{i}(x\curlywedge y)])\in F_{a,b}$. Since $u[\bar {w}|_{i}(x\curlywedge y)]\leq u[\bar{w}|_{i}\,x]$ and $u[\bar{w}|_{i}(x\curlywedge y)]\leq u[\bar{w}|_{i}\,y]$, we have $u[\bar{w}|_{i}(x\curlywedge y)]\leq u[\bar{w}|_{i}\,x]\curlywedge u[\bar{w}|_{i}\,y]$, which, by the stability of ω gives $t_{1}(u[\bar {w}|_{i}(x\curlywedge y)]) \leq t_{1}(u[\bar{w}|_{i}\,x]\curlywedge u[\bar{w}|_{i}\,y])$. Consequently, $t_{1}(u[\bar{w}|_{i}\,x]\curlywedge u[\bar{w}|_{i}\,y])\in F_{a,b}$, so $u[\bar{w}|_{i}\,x]\curlywedge u[\bar{w}|_{i}\,y]\notin W_{a,b}$, i.e., $u[\bar{w}|_{i}\,x]\equiv u[\bar {w}|_{i}\,y](\varepsilon_{a,b})$. In this way we have proved that the relation ε _a,b is i-regular for every i=1,…,n. Thus it is v-regular. □

FormalPara Proposition 11

All equivalence classes of ε _a,b , except of W _a,b , are filters.

FormalPara Proof

Indeed, let H≠W _a,b be an arbitrary class of ε _a,b . If x∈H and x≤y, then x⋏y=x∉W _a,b , consequently, (x,y)∈ε _a,b . Hence, y∈H. Further, let x,y∈H, then (x,y)∈ε _a,b . Thus x⋏y∉W _a,b , i.e., t(x⋏y)∈F _a,b for some t∈T _n (G). But x⋏y=x⋏(x⋏y), hence, t(x⋏(x⋏y))∈F _a,b and x⋏(x⋏y)∉W _a,b . So x≡x⋏y(ε _a,b ). This implies x⋏y∈H. Thus, we have shown that H is a filter. □

FormalPara Proposition 12

If x⋎y exists for some x,y∈W _a,b , then x⋎y∈W _a,b .

FormalPara Proof

Let x⋎y exists for some x,y∈W _a,b . If x⋎y∉W _a,b , then t(x⋎y)∈F _a,b for some t∈T _n (G), and, according to Corollary 2, t(x⋎y)=t(x)⋎t(y). If t(x)∉F _a,b , then F _a,b is a proper subset of the set

$$U =\bigl\{u\in G\,|\,(\exists z\in F_{a,b})\,z\curlywedge t(x)\leq u\bigr\}$$

because t(x)∈U.

We show that U is a filter. 0∉U because, by (15), we have 0≤z⋏t(x) for any z∈F _a,b . Let s∈U and s≤r. Then z⋏t(x)≤s for some z∈F _a,b . Consequently, z⋏t(x)≤r, so r∈U. Now let s∈U and r∈U, i.e., z ₁⋏t(x)≤s and z ₂⋏t(x)≤r for some z ₁,z ₂∈F _a,b . Since F _a,b is a filter, we have z ₁⋏z ₂∈F _a,b . Hence, (z ₁⋏z ₂)⋏t(x)≤s⋏r, which implies s⋏r∈U. Thus U is a filter. But by assumption F _a,b ⊂U is a maximal filter, which does not contain b, so b∈U. Consequently, z ₁⋏t(x)≤b for some z ₁∈F _a,b . Similarly, if t(y)∉F _a,b , then z ₂⋏t(y)≤b for some z ₂∈F _a,b . This implies z⋏t(x)≤b and z⋏t(y)≤b for z=z ₁⋏z ₂. Hence (z⋏t(x))⋎(z⋏t(y)) exists and

$$\bigl(z\curlywedge t(x)\bigr)\curlyvee\bigl(z\curlywedge t(y)\bigr)=z\curlywedge \bigl(t(x)\curlyvee t(y)\bigr) =z\curlywedge t(x\curlyvee y)\in F_{a,b}$$

by (47). But by (50) we have (z⋏t(x))⋎(z⋏t(y))≤b, so z⋏t(x⋎y)≤b. Since z⋏t(x⋎y)∈F _a,b , then, obviously, b∈F _a,b , which is impossible. So, t(x)∈F _a,b or t(y)∈F _a,b , hence x∉W _a,b or y∉W _a,b , contrary to the assumption that x,y∈W _a,b . Thus, the assumption that x⋎y∉W _a,b is incorrect. Therefore x⋎y∈W _a,b . □

6. Each homomorphism of a Menger algebra (G,o) of rank n into a Menger algebra $(\mathcal{F}(A^{n},A),\mathrm{O})$ is called a representation by n-place functions. Thus, $P:G\to \mathcal{F}(A^{n},A)$ is a representation, if

$$P\bigl(x[y_1\ldots y_n]\bigr)=P(x)\bigl[P(y_1)\ldots P(y_n)\bigr]$$

for all x,y ₁,…,y _n ∈G. A representation which is an isomorphism is called faithful (cf. [2–4, 8]). A representation P of (G,o) is a representation of (G,o,−,0) if

$$P(x-y)=P(x)\setminus P(y) \quad {\mathrm{and}} \quad P(0)=\varnothing $$

for all x,y∈G.

Let (P _i ) _i∈I be the family of representations of a subtraction Menger algebra (G,o,−,0) of rank n by n-place functions defined on pairwise disjoint sets (A _i ) _i∈I . By the sum of the family (P _i ) _i∈I we mean the map P:g↦P(g), denoted by ∑ _i∈I P _i , where P(g) is an n-place function on A=⋃ _i∈I A _i defined by P(g)=⋃ _i∈I P _i (g). It is clear (cf. [2, 3]) that P is a representation of (G,o,−,0).

Similarly as in [2, 3] with each determining pair (ε ^∗,W) we can associate the so-called simplest representation $P_{(\varepsilon^{*},W)}$ of (G,o) which assigns to each element g∈G the n-place function $P_{(\varepsilon^{*},W)}(g)$ defined on $\mathcal{H}=\mathcal{H}_{0}\cup\{\{e_{1}\},\ldots,\{e_{n}\}\}$, where $\mathcal{H}_{0}$ is the set of all ε-classes of G different from W such that

$$(H_1,\ldots,H_n,H)\in P_{(\varepsilon,W)}(g)\longleftrightarrow g[H_1\ldots H_n]\subset H,$$

for $(H_{1},\ldots,H_{n})\in\mathcal{H}^{n}_{0}\cup \{(\{e_{1}\},\ldots,\{e_{n}\})\}$ and $H\in\mathcal{H}$.

FormalPara Theorem 2

Each subtraction Menger algebra of rank n is isomorphic to some difference Menger algebra of n-place functions.

FormalPara Proof

Let (G,o,−,0) be a subtraction Menger algebra of rank n. Then the sum

$$P=\sum_{a,b\in G,\,a\nleq b}P_{(\varepsilon^*_{a,b},W_{a,b})}$$

of the family $(P_{(\varepsilon^{*}_{a,b},W_{a,b})} )_{a,b\in G,\,a\nleq b}$ of simplest representations of (G,o) is a representation of (G,o).

Now we show that P is a representation of (G,o,−,0). Let $\mathcal {H}_{0}$ be the set of all ε _a,b -classes of G different from W _a,b . Consider $H_{1},\ldots,H_{n},H\in\mathcal{H}$, where $\mathcal{H}=\mathcal{H}_{0}\cup\{\{e_{1}\},\ldots,\{e_{n}\}\}$, such that $(H_{1},\ldots,H_{n},H)\in P_{(\varepsilon^{*}_{a,b},W_{a,b})}(g_{1}-g_{2})$ for some g ₁,g ₂∈G. Then, obviously, (g ₁−g ₂)[H ₁…H _n ]⊂H≠W _a,b . Thus $(g_{1}-g_{2})[\bar{x}]\in H$ for each $\bar{x}\in H_{1}\times\cdots\times H_{n}$, which, by (11), gives $g_{1}[\bar{x}]-g_{2}[\bar{x}]\in H$. But $g_{1}[\bar{x}]-g_{2}[\bar {x}]\leq g_{1}[\bar{x}]$ and H is a filter (Proposition 11), hence $g_{1}[\bar{x}]\in H$. Thus $(g_{1}[\bar{x}]-g_{2}[\bar{x}])\curlywedge g_{2}[\bar{x}]=0$, by (33). Consequently, $(g_{1}[\bar{x}]-g_{2}[\bar{x}])\curlywedge g_{2}[\bar{x}]\in W_{a,b}$, because the other ε _a,b -classes as filters do not contain 0. This means that $g_{1}[\bar{x}]-g_{2}[\bar{x}]\not\equiv g_{2}[\bar {x}](\varepsilon_{a,b})$. Hence, $g_{2}[\bar{x}]\notin H$. Therefore g ₁[H ₁…H _n ]⊂H and g ₂[h ₁…H _n ]∩H=∅, which implies

$$(H_1,\ldots,H_n,H)\in P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1)\setminus P_{(\varepsilon^*_{a,b},W_{a,b})}(g_2).$$

In this way, we have proved the inclusion

$$ P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1-g_2)\subset P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1)\setminus P_{(\varepsilon^*_{a,b},W_{a,b})}(g_2).$$

(64)

To show the reverse inclusion let

$$(H_1,\ldots,H_n,H)\in P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1)\setminus P_{(\varepsilon^*_{a,b},W_{a,b})}(g_2).$$

Then $(H_{1},\ldots,H_{n},H)\in P_{(\varepsilon^{*}_{a,b},W_{a,b})}(g_{1})$ and $(H_{1},\ldots,H_{n},H)\notin P_{(\varepsilon^{*}_{a,b},W_{a,b})}(g_{2})$, i.e., g ₁[H ₁…H _n ]⊂H and g ₂[H ₁…H _n ]∩H=∅. Thus $g_{1}[\bar{x}]\in H$ and $g_{2}[\bar{x}]\notin H$ for all $\bar{x}\in H_{1}\times\cdots\times H_{n}$. Since from $g_{1}[\bar {x}]\curlywedge g_{2}[\bar{x}]\notin W_{a,b}$, it follows $g_{1}[\bar {x}]\equiv g_{2}[\bar{x}](\varepsilon_{a,b})$ and $g_{2}[\bar{x}]\in H$, which is a contradiction, we conclude that $g_{1}[\bar{x}]\curlywedge g_{2}[\bar{x}]\in W_{a,b}$.

If $g_{1}[\bar{x}]-g_{2}[\bar{x}]\in W_{a,b}$, then, by (53) and Proposition 12, we obtain $g_{1}[\bar{x}]=(g_{1}[\bar{x}]\curlywedge g_{2}[\bar{x}])\curlyvee(g_{1}[\bar {x}]-g_{2}[\bar{x}])\in W_{a,b}$. Consequently, $g_{1}[\bar{x}]\in W_{a,b}$, which is impossible because $g_{1}[\bar{x}]\in H$. Thus, $(g_{1}[\bar{x}]-g_{2}[\bar{x}])\curlywedge g_{1}[\bar{x}]=g_{1}[\bar{x}]-g_{2}[\bar{x}]\notin W_{a,b}$. Hence, $g_{1}[\bar {x}]-g_{2}[\bar{x}]\equiv g_{1}[\bar{x}](\varepsilon_{a,b})$. This implies $(g_{1}-g_{2})[\bar{x}]=g_{1}[\bar{x}]-g_{2}[\bar{x}]\in H$. Therefore, (g ₁−g ₂)[H ₁…H _n ]⊂H, i.e., $(H_{1},\ldots,H_{n},H)\in P_{(\varepsilon^{*}_{a,b},W_{a,b})}(g_{1}-g_{2})$. So, we have proved

$$P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1)\setminus P_{(\varepsilon^*_{a,b},W_{a,b})}(g_2)\subset P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1-g_2).$$

This together with (64) proves

$$P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1-g_2)= P_{(\varepsilon^*_{a,b},W_{a,b})}(g_1)\setminus P_{(\varepsilon^*_{a,b},W_{a,b})}(g_2),$$

which means that P(g ₁−g ₂)=P(g ₁)∖P(g ₂) for g ₁,g ₂∈G. Further, P(0)=P(0−0)=P(0)∖P(0)=∅. So, P is a representation of (G,o,−,0) by n-place functions.

We show that this representation is faithful. Let P(g ₁)=P(g ₂) for some g ₁,g ₂∈G. If g ₁≠g ₂, then both inequalities g ₁≤g ₂ and g ₂≤g ₁ at the same time are impossible. Suppose that g ₁≰g ₂. Then $g_{1}\in F_{g_{1},g_{2}}$ and, consequently,

$$\bigl(\{e_1\},\ldots,\{e_n\},F_{g_1,g_2}\bigr)\in P_{(\varepsilon^*_{g_1,g_2},W_{g_1,g_2})} (g_2).$$

Since $P_{(\varepsilon^{*}_{g_{1},g_{2}},W_{g_{1},g_{2}})}(g_{1})=P_{(\varepsilon ^{*}_{g_{1},g_{2}},W_{g_{1},g_{2}})}(g_{2})$, then, obviously,

$$\bigl(\{e_1\},\ldots,\{e_n\},F_{g_1,g_2}\bigr)\in P_{(\varepsilon^*_{g_1,g_2},W_{g_1,g_2})} (g_2).$$

Thus $\{g_{2}\}=g_{2}[\{e_{1}\}\ldots\{e_{n}\}]\subset F_{g_{1},g_{2}}$, hence $g_{2}\in F_{g_{1},g_{2}}$. This is a contradiction because $F_{g_{1},g_{2}}$ is a filter containing g ₁ but not containing g ₂. The case g ₂≰g ₁ is analogous. So, the supposition g ₁≠g ₂ is not true. Hence g ₁=g ₂ and P is a faithful representation. The theorem is proved. □