Ambiguous Representations of Semilattices, Imperfect Information, and Predicate Transformers

Crisp and lattice-valued ambiguous representations of one continuous semilattice in another one are introduced and operation of taking pseudo-inverse of the above relations is defined. It is shown that continuous semilattices and their ambiguous representations, for which taking pseudo-inverse is involutive, form categories. Self-dualities and contravariant equivalences for these categories are obtained. Possible interpretations and applications to processing of imperfect information are discussed.


Introduction
The goal of this work is to generalize notions of crisp and L-fuzzy ambiguous representations introduced in [15] for closed sets in compact Hausdorff spaces.Fuzziness and roughness were combined to express the main idea that a set in one space can be represented with a set in another space, e.g., a 2D photo can represent 3D object.This representation is not necessarily unique, and the object cannot be recovered uniquely, hence we say "ambiguous representation".
It turned out that most of results of [15] can be extended to wider settings, namely to continuous semilattices, which are standard tool to represent partial information in denotational semantics of programming languages.Consider a computational process or a system which can be in different states, and these states can change, e.g., a game, position in which changes after moves of players.Assume that any party involved (e.g., a player) at a moment of time can obtain only imperfect information about the state of the system ("imperfect" means that this information only reduces uncertainty but not necessarily eliminates it).All possible portions of information we can obtain in an observation form a domain of computation S [3], which is usually required to be a continuous meet semilattice with zero (cf.[17] for a detailed explanation why continuous, semilattices are an appropriate tool for this purpose, and [5,9] for more information on continuous posets).If x y in S, then an information (a statement) x is more specific (restrictive) than y.Respectively 0 ∈ S means "no information".The meet of x and y is the most specific piece of information including both x and y (as particular cases).It is not necessarily equivalent to "x or y" in the usual logical sense.
Example 2 Let S be the hyperspace exp X of all non-empty closed sets of a compactum X.
Points of X are possible states of the system, and A ⊂ cl X represents the fact that one of the states x ∈ A is achieved.Then exp X is ordered by reverse inclusion, hence X ∈ exp X is the least element that means "anything can happen".The meet of A and B is their union, therefore can be interpreted as "A or B".The ambiguous representations were first introduced and their properties proved for this particular case in [15].Domain of computation is the primary interpretation of a continuous semilattice in this paper.We use monotonic binary-or lattice-valued predicates [8] on semilattices to express degrees of belief that certain pieces of information describe actual state of a system or of a process.
The goal of this work is to study relations between imperfect information on the same system or process obtained in different observation, e.g., how information changes after a step of a computational process or after a move in the game.We consider the introduced ambiguous representations to be an adequate tool for this task.
The paper is organized as follows.First necessary definitions and facts are given on continuous (semi-)lattices and monotone predicates.Then we define compatibilities, which are functions with values 0 and 1 that show whether two pieces of information can be valid together.Next ambiguous representations are introduced as crisp and L-fuzzy relations between continuous semilattices.Operation of taking pseudo-inverse is defined for these relations, its properties are proved, and classes of pseudo-invertible representations are investigated.It is shown that continuous semilattices and their pseudo-invertible crisp and L-fuzzy ambiguous representations form categories, and can be equivalently regarded as isotone mappings or linear operators between idempotent semimodules.Self-dualities and contravariant equivalences are constructed for these categories.We also discuss possible applications of the developed theory.

Preliminaries
We adopt the following definitions and notation, which are consistent with [5,10].Proofs of the facts below can also be found there.From now on, semilattice means meet semilattice, if otherwise is not specified.If a poset contains a bottom (a top) element, then it is denoted by 0 (resp.by 1).A top (a bottom) element in a semilattice is also called a unit (resp.a zero).
Order (2020) 37:319-339 For a partial order on a set X, the relation ˜ , defined as x ˜ y ⇐⇒ y x, for x, y ∈ X, is a partial order called opposite to , and (X, ) op denotes the poset (X, ˜ ).If the original order is obvious, we write simply X op for the reversed poset.We also apply ( ) to all notation to denote passing to the opposite order, i.e. write X = X op , sup = inf, 0 = 1 etc.For a morphism f : (X ) → (Y, ) in a category Poset of posets and isotone (order preserving) mappings, let f op be the same mapping, but regarded as (X, ˜ ) → (Y, ˜ ).It is obvious that f op is isotone as well, thus a functor (−) op : Poset → Poset is obtained.For a subset A of a poset (X, ), we denote A↑ = {x ∈ X | a x for some a ∈ A}, A↓ = {x ∈ X | x a for some a ∈ A}.

lower).
A topological meet (or join) semilattice is a semilattice L carrying a topology such that the mapping ∧ : A poset (X, ) is called complete if each non-empty subset A ⊂ X has a least upper bound.
A set A in a poset (X, ) is directed (filtered) if, for all x, y ∈ A, there is z ∈ A such that x z, y z (resp.z x, z y).A poset is called directed complete (dcpo for short) if it has lowest upper bounds for all its directed subsets.
The Scott topology σ (X) on (X, ) consists of all those U ⊆ X that satisfy x ∈ U ⇔ U ∩ D = ∅ for every −directed D ⊆ X with a least upper bound x.Note that "⇐" above implies U = U ↑.
In a dcpo X, a set is Scott closed iff it is lower and closed under suprema of directed subsets.
A mapping f between dcpo's X and Y is Scott continuous, i.e. continuous w.r.t.σ (X) and σ (Y ), if and only if it preserves suprema of directed sets.
Let L be a poset.We say that x is way below y and write x y iff, for all directed subsets D ⊆ L such that sup D exists, the relation y ≤ sup D implies the existence of d ∈ D such that x ≤ d. "Way-below" relation is transitive and antisymmetric.An element satisfying x x is said to be compact or isolated from below, and in this case the set {x}↑ is Scott open.
A poset L is called continuous if, for each element y ∈ L, the set y↓ ↓ = {x ∈ L | x y} is directed and its least upper bound is y.A domain is a continuous dcpo.If a domain is a semilattice, it is called a continuous semilattice.
A complete lattice L is called completely distributive if, for each collection of sets holds.This property implies distributivity, but the converse fails.Then both L and L op are continuous, and the join of Scott topologies on L and L op provides the unique compact Hausdorff topology on L with a basis consisting of small sublattices (Lawson topology).In the sequel completely distributive lattices will be regarded with Lawson topologies. For Order (2020) 37:319-339 The following obvious property is quite useful.
Proof Necessity is obvious.To prove sufficiency, observe that all cuts of R being Scott closed (hence lower sets) implies that R is a lower set as well.Without loss of generality we can consider only the case n = 2. Let a subset D ⊂ R ⊂ S 1 × S 2 be directed.Then the set D = D↓ ⊂ R is directed as well and lower in S 1 × S 2 , hence is the product D 1 × D 2 of directed lower sets D 1 ⊂ S 1 and D 2 ⊂ S 2 .For any x ∈ D 1 the set {x} × D 2 is contained in R, therefore D 2 is contained in the Scott closed cut xR.This implies that the least upper bound b = sup D 2 , which exists because S 2 is a dcpo, is an element of this cut, hence (x, b) ∈ R for all x ∈ D 1 .Thus D 1 is contained in the Scott closed cut Rb, therefore a = sup D 1 in the dcpo S 1 also belongs to this cut.We obtain that (a, b)

, n}, and subsets
In the sequel L will be a completely distributive lattice [5], which will be used to keep truth values ("degrees of belief").Its top element 1 means "surely true", and the bottom element 0 means "quite impossible".In the simplest case L = 2 = {0, 1}, i.e., we obtain binary logic "yes/no".
In Let Sem ↑ be the category of all continuous semilattices and all Scott continuous mappings, and Sem 0 be its subcategory consisting of all continuous semilattices with zeros and all Scott continuous zero-preserving semilattice morphisms.
Following [8], for a semilattice S we call the elements of the set M [L] S = [S L op ] op L-fuzzy monotonic predicates on S. For m ∈ [S L op ] op and a ∈ S, we regard m(a) as the truth value of a, hence it is required that m(b) m(a) for all a b.The second op means that we order the fuzzy predicates pointwise, i.e. m 1 m 2 iff m 1 (a) m 2 (a) in L (not in L op !) for all a ∈ S. For S with a least element 0, consider also the subset S of all normalized predicates that take 0 ∈ S (no information) to 1 ∈ L (complete truth).It is natural to require m(0) = 1 because we cannot be wrong if we say nothing.Observe that In particular, a binary monotonic predicate m ∈ M [2] S takes each piece a of information to 1 if it is true or to 0 if it is wrong.Such predicate is completely determined with the Scott continuous set m −1 (1) ⊂ S of true statements.On the other hand, each Scott closed set F ⊂ S determines the binary monotonic predicate It follows from [4, Theorem 4] (although called "folklore knowledge" in [8]) that, for a domain D and a completely distributive lattice L, the set [D L op ] is a completely distributive lattice, hence this is also valid for M [L] S, where S is a continuous semilattice.If S possesses a least element, then M [L] D is a completely distributive lattice as well.Infima in M [L] S and M [L] S are calculated pointwise, but suprema need "adjustment"

Compatibilities for Continuous Semilattices
We use the results of [13] and denote by S ∧ the set of all (probably empty) Scott open filters in a continuous semilattice S with zero except S itself.We order S ∧ by inclusion, then S ∧ is a continuous semilattice with the bottom element ∅.Then S ∧ can be regarded as [S → {0, 1}] 0 , i.e., its elements can be identified with the bottom-preserving meetpreserving Scott continuous maps S → {0, 1} (the preimages of {1} under such maps are precisely the non-trivial Scott open filters in S).For an arrow f : , is a self-duality.In fact it is a restriction of the Lawson duality [5].
We slightly change the terminology introduced in [13]: Definition 1 Let S, S be continuous semilattices with bottom elements respectively 0, 0 .A mapping P : S × S → {0, 1} is called a compatibility if: (1) P is meet preserving in the both variables, and P (0, y) = P (x, 0 ) = 0 for all x ∈ S, y ∈ S ; (2) P is Scott continuous.
If, additionally, the following holds: (3) P separates elements of S and of S , i.e.: (3a) for each x 1 , x 2 ∈ S, if P (x 1 , y) = P (x 2 , y) for all y ∈ S , then x 1 = x 2 ; (3b) for each y 1 , y 2 ∈ S , if P (x, y 1 ) = P (x, y 2 ) for all x ∈ S, then y 1 = y 2 ; then we call P a separating compatibility.
The definition of (separating) compatibility is symmetric in the sense that the mapping P : S × S → {0, 1}, P (y, x) = P (x, y) is a (separating) compatibility as well, which we call the reverse compatibility.For compatibilities we use also infix notation xP y ≡ P (x, y).
We can consider a compatibility P : S × S → {0, 1} as a characteristic mapping of a binary relation We interpret P (x, y) = 1 as "pieces x and y of information are incompatible (cannot be valid simultaneously)", hence probably a longer term "incompatibility" would be more adequate.Meet-preservation in the first argument means that if x 1 and x 2 are incompatible (cannot be valid together) with y, then "x 1 or x 2 " is incompatible with y as well, similarly for the second argument.A compatibility P : S ×S → {0, 1} is separating if for all x 1 = x 2 in S there is y ∈ S such that exactly one of x i is incompatible with y w.r.t.P , similarly for y 1 = y 2 in S and x ∈ S. Then elements of S can be regarded as "negative statements" about state of the system observed at S : given y ∈ S and a separating compatibility P , we declare impossible all x ∈ S such that (x, y)P 1.
Example 3 Let L be a completely distributive lattice, then so is L = L op .The mapping P : L × L op → {0, 1} defined as is a separating compatibility.
The following statement from [13] is of crucial importance: Proposition 1 Let S, S be continuous meet semilattices with bottom elements 0, 0 resp.If P : S × S → {0, 1} is a separating compatibility, then the mapping i that takes each x ∈ S to xP is an isomorphism S → S ∧ .Conversely, each isomorphism i : S → S ∧ is determined by the above formula for a unique separating compatibility P : S ×S → {0, 1}.
In particular, this together with the above example implies that L ∧ ∼ = L op for a completely distributive lattice L.
For a fixed separating compatibility P : S × S → {0, 1} and subsets A ⊂ S, B ⊂ S , the sets will be called the transversals of A and B respectively.In other words, B ⊥ is the set of all statements in S that are compatible with all "negative statements" from B ⊂ S , similarly for A ⊥ .Hence elements of S prevent or prohibit elements of S, and vice versa.Separation means that each element is uniquely determined with all the elements it prohibits.
It is easy to see that A ⊥ and B ⊥ are Scott closed, and A ⊥⊥ = (A ⊥ ) ⊥ is the Scott closure of A = ∅, i.e., the least Scott closed (hence lower) subset in S that contains A, similarly for B ⊥⊥ .
Obviously the transversal operation (−) ⊥ is antitone, i.e., A ⊂ B implies A ⊥ ⊃ B ⊥ , and for a filtered family {A α | α ∈ A} of closed lower sets the equality

Category of Ambiguous Representations
We call such ambiguous representations crisp to distinguish them from L-fuzzy ambiguous representations that will be defined in the next section.
Recall that, for an ambiguous representation R ⊂ S 1 ×S 2 and an element x ∈ S 1 , the nonempty Scott closed set xR ⊂ S 2 determines the normalized binary monotonic predicate The correspondence x → m xR , which we also denote with R, is isotone.Conversely, each isotone mapping R : Therefore in the sequel we can equivalently regard ambiguous representations either as relations or as isotone mappings.For each ambiguous representation R of S 1 in S 2 we use notation R : We interpret xRy as "if x is true, then y can be (but not necessarily is) true" or "y is attainable from x".Hence xR is the set of all y attainable from x (under R), and m xR is the binary monotonic predicate that selects all elements of S 2 that can be true provided x ∈ S 1 is true.
For an ambiguous representation R : We explain why such definition has been chosen.If xRy and yP ŷ = 1, then ŷ is incompatible with some "consequence" of x, hence ŷ excludes x.If all the elements x ∈ S 1 incompatible with x ∈ Ŝ1 are among those excluded by ŷ, then a "negative statement" x can be considered a "consequence" of a "negative statement" ŷ, which we denote ŷR x.
It is easy to observe that 02 R = { 01 }, because 02 ∈ Ŝ2 excludes nothing in S 1 , therefore 01 ∈ Ŝ1 is the only possible "consequence" of 02 .
Obviously R is an ambiguous representation Ŝ2 ⇒ Ŝ1 as well, hence we can calculate R = (R ) : S 1 ⇒ S 2 using the reverse (in fact the same) separating compatibilities again.

Proposition 2 For each ambiguous representation
Proof Necessity of (d) has been already explained.Assume it holds.Observe the validity of the formula Taking into account for all x ∈ S 1 , which is the required inclusion.
The set {x ∈ S 1 | ŷ ∈ (xR) ⊥ } is lower, and by (d) is non-empty, therefore its double transversal is the closure, thus The Let R be pseudo-invertible, hence satisfy (c).To show Scott continuity of R : x 0 } for all x 0 ∈ S 1 , which in fact means that x 0 R = Cl( x x 0 xR).If y 0 ∈ x 0 R, then for all y y 0 by (c) there is x x 0 such that xRy, hence y ∈ x x 0 xR.Taking into account y 0 ∈ Cl{y ∈ S 2 | y y 0 }, we obtain x 0 R ⊂ Cl( x x 0 xR).The reverse inclusion is immediate.Now let R : S 1 → M [2] S 2 be Scott continuous, and x 0 Ry 0 .Then the limit of the directed net of binary monotonic predicates m xR , x x 0 , i.e., their lowest upper bound, is equal to m x 0 R , hence inf sup i.e., for all y y 0 there is x x 0 such that m xR (y) = 1 ⇐⇒ xRy.Therefore (c) is valid, and, if (d) is valid as well, then R is pseudo-invertible.
One of the reasons to consider this subclass is that, if we compose ambiguous representations as relations, i.e., for then the resulting relation can fail to satisfy closedness in the condition (b) of the definition of ambiguous representation, hence ambiguous representations do not form a category.
To improve things, redefine the composition as Now closedness is at hand, but the composition ";" is not associative.

Corollary 1 Let ambiguous representations
and the composition R;Q is pseudo-invertible as well.
Proof By the above and taking into account that (−) is isotone:

Proposition 4 Composition ";" of the pseudo-invertible ambiguous representations is associative.
Proof Recall that, for ambiguous representations R ⊂ S 1 × S 2 , Q ⊂ S 2 × S 3 , the composition is calculated as It is also important that, for elements a c in a continuous semilattice, there is an element b such that a b c.Now, let (x, t) ∈ R;(Q;T ).For any t t choose t such that t t t, then there is y ∈ S 2 such that (x, y) ∈ R, (y, t ) ∈ Q;T .The latter implies that there is z ∈ S 3 such that (y, z) ∈ Q, (z, t ) ∈ T .
Similarly, let (x, t) ∈ (R;Q);T , then for all t t choose t t t, and there is z ∈ S 3 such that (x, z ) ∈ R;Q, (z , t ) ∈ T .Pseudo-invertibility of T implies the existence of z z such that (z, t ) ∈ T as well.There is also y ∈ S 2 , (x, y) ∈ R, (y, z) ∈ Q.On the other hand, if, for x ∈ S 1 , t ∈ S 4 , elements y ∈ S 2 , z ∈ S 3 exist for all t t such that (x, y) ∈ R, (y, z) ∈ Q, (z, t ) ∈ T , then (x, t ) ∈ RQT , hence (x, t ) ∈ R(Q;T ) and (x, t ) ∈ (R;Q)T , which in turn implies both (x, t) ∈ R;(Q;T ) and (x, t) It is easy to verify that, for a continuous semilattice S with zero, the relation x} is a pseudo-invertible ambiguous representation that is a neutral element for composition.Thus: Proposition 5 All continuous semilattices with bottom elements and all pseudo-invertible ambiguous representation form a category SemPR, which contain Sem 0 as a subcategory.
An obvious embedding Sem 0 → SemPR is of the form: I S = S for an object S, and We denote an arrow R from S 1 to S 2 in SemPR with R : S 1 ⇒ S 2 and use for the composition of R and Q the synonymic notations R;Q (in direct order) and Q • R (in reverse order) both in Sem 0 and SemPR.

Category of L-Fuzzy Ambiguous Representations
Now we extend the notion of ambiguous representation to lattice-valued relations in the spirit of [12] and [13].

Definition 4
Let S 1 , S 2 be continuous semilattices with zeros 0 1 and 0 2 resp., L a completely distributive lattice with a bottom element 0 and a top element 1.An L-fuzzy ambiguous representation (or an L-ambiguous representation for short) of S 1 in S 2 is a ternary relation R ⊂ S 1 × S 2 × L such that (a) if (x, y, α) ∈ R, x x in S 1 , y y in S 2 , and α α in L, then (x , y , α ) ∈ R as well; and contains all the elements of the forms (0 2 , α) and (y, 0); (c) for all x ∈ S 1 , y ∈ S 2 , α, β ∈ L, if (x, y, α) ∈ R, (x, y, β) ∈ R, then (x, y, α∨β) ∈ R.
The following definition is equivalent.Obviously, due to complete distributivity of L, (c ) is equivalent to any of the following properties: (c ) for all x ∈ S 1 y ∈ S 2 the set xyR = {α ∈ L | (x, y, α) ∈ R} is a non-empty directed lower set such that, if β ∈ xyR for all β α, then α ∈ xyR; or (c ) for all x ∈ S 1 , y ∈ S 2 the set xyR is a lower set with a greatest element (i.e., a set of the form {α}↓).
and these conversions are mutually inverse.Hence we can equivalently regard L-ambiguous representations either as relations or as isotone mappings, whatever is more convenient.An L-ambiguous representation of S 1 in S 2 is denoted with R : S 1 ⇒ L S 2 .In this paper we interpret (x, y, α) as "if x is true, then the truth value of y is at least α" or "given x, y is attainable with degree of belief at least α".By definition, for all x and y there is the greatest such α, which can also be treated as quality/precision of representation of one piece of information with another one.
Observe also that (a )+(b ) mean that, for all α ∈ L, the cut Rα = {(x, y) ∈ S 1 × S 2 | (x, y, α) ∈ R} is a (crisp) ambiguous representation of S 1 in S 2 as defined in the previous section.We will call it the α-cut of R and denote R α .
For an ambiguous representation R ⊂ S 1 × S 2 × L define the relation R ⊂ Ŝ2 × Ŝ1 × L through its α-cuts as follows: or, equivalently, with the formulae A shorter formula uses transversals:

Proposition 7
The relation R is an L-ambiguous representation as well.
Proof For the intersection of crisp ambiguous representations is a crisp ambiguous representation as well, (a )+(b ) for R are immediate.To verify c , assume that β ∈ ŷ xR for all β α = 0, i.e., ( ŷ, x) ∈ β α γ β (R γ ) .In a completely distributive lattice L we have γ α if and only if there is Obviously ŷ xR is a lower set that contains 0. Show that it is directed.If α, β ∈ ŷ xR , then for all γ α ∨ β there are α α, β β such that γ α ∨ β .Therefore Thus α ∨ β ∈ ŷ xR , and the latter set is directed in L.
Hence R = (R ) ⊂ S 1 × S 2 × L is an L-ambiguous representation as well.

Proposition 8 For each ambiguous representation
Proof Necessity of (e) is an immediate corollary of the analogous property (d) for the pseudo-inverses of crisp ambiguous representations.Assume that (e) holds.Recall that Order (2020) 37:319-339 Hence (double transversal of the non-empty by (e) set is its Scott closure) (Scott closure of a lower set A consists of all points approximated by elements of A) therefore R ⊂ R. Clarify why the "=" sign with an asterisk is valid.Obviously, if γ β and ŷ ∈ (x R γ ) ⊥ , then ŷ ∈ (x R β ) ⊥ , and (−) ⊥ is antitone, hence "⊃" is immediate.On the other hand, there is γ β such that for all x x ŷ ∈ (x R γ ) ⊥ =⇒ for all x x there is γ therefore "⊂" (after renaming γ with β) is also obtained.It is also clear that, for R = R to be valid, it is necessary and sufficient that the only "⊂" in the above sequence is "=", i.e., each y ∈ xR α must be a closure point of all lower sets {x}↓ ↓ R β for β α, which in fact is (d).
The L-ambiguous representations R that satisfy R = R are also called pseudoinvertible.Similarly to the binary case:

invertible if and only if, when regarded as a mapping S 1 → M [L] S 2 , it is Scott continuous and bottompreserving.
Proof is also obtained mutatis mutandis.
To define composition of L-ambiguous representations, we need an additional operation * : L × L → L that makes L = (L, * ) a unital quantale, i.e., this operation is infinitely distributive w.r.t.supremum in both variables (hence Scott continuous) and 1 is a two sided unit for " * ".Note that commutativity is not demanded, hence from now on " * " is a (possibly) noncommutative lower semicontinuous conjunction for an L-valued fuzzy logic [2,7].The operation α * β ≡ β * α satisfies the same requirements, hence L = (L, * ) is a unital quantale as well.
Then, for L-ambiguous representations R ⊂ S 1 × S 2 × L and Q ⊂ S 2 × S 3 × L, the composition R * Q can be defined in a manner usual for L-fuzzy relations: Similarly to the case of crisp ambiguous representations, this composition is associative, but R * Q can fail to be an L-ambiguous representation, namely (b) is not always valid.Therefore we "improve" the composition as follows: R * Here is an expanded version of the latter definition: for x ∈ S 1 , z ∈ S 3 , α ∈ L we have (x, z, α) ∈ R * ,Q if and only if for all z z, α α there are n ∈ N, y 1 , . . ., y n ∈ S 2 , β 1 , γ 1 , . . ., β n , γ n ∈ L such that The simplest operation " * " that obviously satisfies the above conditions is the lattice meet "∧".In this case for " * ," we use the denotation ";".For an L-ambiguous representation R we regard R as a L-ambiguous representation, and use " * ," for the compositions of such representations.
Proof is similar to the one for crisp representations and reduces to the observation that, for ) are equivalent to the existence, for all α α and t t, Hence we obtain a category:

Proposition 11 All continuous semilattices with bottom elements and all pseudo-invertible L-ambiguous representations form a category SemPR *
L , which contains SemPR as a subcategory.
The embedding I * L : SemPR → SemPR * L preserves the objects and turns each crisp ambiguous representation R ⊂ S 1 × S 2 into an L-ambiguous one as follows: Clearly there is also the embedding I * L : SemPR → SemPR * L into the category built upon the "swapped" operation " * ".
We denote an arrow R from S 1 to S 2 with R : S

Ambiguous Representations as Affine Operators and Predicate Transformers
Observe that compositions of (crisp or L-fuzzy) ambiguous representations were calculated only for their "relational" forms, but not for them given as isotone mappings.Interpretation of "slicewise defined" pseudo-inverses to L-ambiguous representations is also unclear.To fill this gap, we briefly present results of [14] and show the connection with the developed theory.
Recall that we treat * as a (possibly noncommutative) conjunction in an L-valued fuzzy logic, and ⊕ = ∨ will be a disjunction.The Boolean case is obtained for L = {0, 1} and * = ∧.
A (left idempotent) (L, ⊕, * )-semimodule [1] (or L-semimodule for short if the operations ⊕ and * are clearly understood) is a set X with operations ⊕ : X × X → X and * : L × X → X such that for all x, y, z ∈ X, α, β ∈ L : (1) x ⊕ y = y ⊕ x; (2) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z); (3) there is an (obviously unique) element 0 ∈ X such that x ⊕ 0 = x for all x; Observe that these axioms imply that (X, ⊕) is an upper semilattice with a bottom element 0, the order is defined as x y ⇐⇒ x ⊕y = y, and α * 0 = 0 for all α ∈ L. The operation * is isotone in both variables.Hence an (L, ⊕, * )-semimodule is an analogue of a vector space.Similarly, analogues exist for linear and affine mappings.A mapping f : X → Y between (L, ⊕, * )semimodules is called linear if, for all x 1 , . . ., x n ∈ X and α 1 , . . ., α n ∈ L, the equality If the latter equality is ensured only whenever α 1 ⊕ . . .⊕ α n = 1, then f is called affine.Observe that an affine mapping f preserves joins, i.e. f (x 1 ⊕ x 2 ) = f (x 1 ) ⊕ f (x 2 ) for all x 1 , x 2 ∈ X.An affine mapping is linear if and only if it preserves the least element.
Observe that such (X, ⊕) has a least element, a greatest element, and suprema for all subsets, therefore is a continuous lattice.If the poset (X, ⊕) is a completely distributive lattice, then we call (X, ⊕, * ) a completely distributive (L, ⊕, * )-semimodule.It was proved [14] that, for a domain D with a bottom element, (M [L] D, ⊕, ¯ ) is a completely distributive (L, ⊕, * )-semimodule with the operations The required extension is determined by the formula This means that M [L] D is a free object in the category (L, ⊕, * )-CSMod ↑ of continuous L-semimodules and Scott continuous linear mappings over an object D of the category Dom ↑ of domains with bottom elements and their Scott continuous bottom-preserving mappings.
An immediate corollary (cf.[14], proof of Proposition 2.3) is that, for domains D 1 , D 2 with bottom elements, each Scott continuous linear mapping Φ : Recall that, for continuous semilattices S 1 , S 2 with a Scott continuous bottom-preserving mapping S 1 → M [L] S 2 is precisely a pseudo-invertible L-ambiguous representation.
Thus we arrive at:

Proposition 15 The correspondence that assigns M [L] S to each continuous semilattice S and the unique Scott continuous linear extension M
1 is a functor that embeds the category SemPR * L into the category (L, ⊕, * )-CSMod ↑ of continuous L-semimodules as a full subcategory.
In fact the latter subcategory is the Kleisli category for the respective monad [10], but we will not go into detail here.Now it is obvious how to compose L-ambiguous representation in "functional" form: first extend them to linear mappings of free semimodules, then compose in (L, ⊕, * )-CSMod ↑ , and, finally, restrict back to semigroups.Now it is time to present practical sense of the constructed extensions, following [16].For a domain of computation D, we treat each mapping m : D → L as "it is known that, for each d ∈ D, its truth value is (or can be) at least m(d)".Similarly, an arbitrary mapping ϕ : D → M [L] D is interpreted as "if a ∈ D is true, then the truth value of each b ∈ D is at least ϕ(a)(b)".Note that ϕ(a)(b) is implicitly considered as a "conditional" truth value, i.e., if a is "partially true" at a degree α, then b is true at least at a degree α * ϕ(a)(b).
Hence, such a ϕ is an L-fuzzy state transformer.The same formula is valid also for normalized predicates, and it is easy to observe that the strongest postcondition predicate transformer Φ is exactly the unique linear Scott continuous extension of the state transformer ϕ.

Pseudo-Inverses as Hermitian Conjugates
To clarify what the pseudo-inverse to an ambiguous representation is, we use a "scalar-like" product of monotonic predicates, which was introduced in [14].
Let S be a continuous semilattice with bottom elements, and P : S × S → {0, 1} be a separating compatibility.We regard M [L] S as a continuous L-semimodule and M [L] Ŝ as a continuous L-semimodule, where L is L with the "swapped" multiplication.P is precisely the supremum of such "incompatibilities" m(s) * m (s ), for sP s = 1, hence it can be regarded as a measure of total incompatibility of m and m .
The following notion was also introduced in [14].Assume that K 1 , K 2 are Lsemimodules, K 1 , K 2 are L-semimodules, multiplications • : K 1 × K 1 → L and • : K 2 × K 2 → L are such that K 1 , K 1 and K 2 , K 2 are dual pairs, and A : K 1 → K 2 is a linear mapping.It is natural to call a linear mapping A : K 2 → K 1 the (Hermitian) conjugate to A if Aa • a = a • A a whenever a ∈ K 1 , a ∈ K 2 .The separation property implies that, if a conjugate for a given A exists, it is unique.Hence we write A = A * in this case, and obviously A * * = A. It is also immediate that, for the composition A • B of linear mappings with conjugates A * and B * , respectively, a conjugate exists and is equal to B * • A * .Order (2020) 37:319-339 the sequel D D denotes a Scott continuous mapping between domains D and D , and [D D ] is the set of all such mappings.If S and S are continuous (meet) semilattices, then a Scott continuous meet preserving mapping is denoted with S → S , with [S → S ] being a notation for the respective sets of mappings.If the above posets have zero (bottom) elements, then [D D ] 0 and [S → S ] 0 are the subsets consisting of the bottom-preserving mappings.

Proposition 3
family {{xR) ⊥ | x x} of closed lower sets is filtered, hence the transversal of its intersection equals Cl {(xR) ⊥⊥ | x x} = Cl {xR | x x} , and the equality R = R is equivalent to xR = Cl {xR | x x} , which in fact is the condition (c).The equality R = R implies (R ) = R , hence, if (c), (d) hold for R, then they hold for R as well.Therefore on such ambiguous representations the operation ( ) is involutive, and we call each of R and R the pseudo-inverse to the other one.The ambiguous representations satisfying (c), (d) are called pseudo-invertible.An ambiguous representation R ⊂ S 1 × S 2 is pseudo-invertible if and only if, when regarded as a mapping S 1 → M [2] S 2 , it is Scott continuous and bottom-preserving.Proof Clearly (d) is equivalent to R(0 1 ) = m {0 2 } , and m {0 2 } is the least element of M [2] S 2 , thus (d) means that R is bottom-preserving.

Definition 5
For continuous semilattices S 1 , S 2 with zeros and a completely distributive lattice L, a ternary relation R ⊂ S 1 × S 2 × L is an L-ambiguous representation of S 1 in S 2 if (a ) for all y ∈ S 2 , α ∈ L the set Ryα = {x ∈ S 1 | (x, y, α) ∈ R} is an upper set in S 1 ; (b ) for all x ∈ S 1 , α ∈ L the set xRα = {y ∈ S 2 | (x, y, α) ∈ R} is non-empty and Scott closed in S 2 ; (c ) for all x ∈ S 1 , y ∈ S 2 the set xyR = {α ∈ L | (x, y, α) ∈ R} is non-empty, directed, and Scott closed in L.

Fix x ∈ S 1 ,
then due to (c ) the formula m xR (y) = max{α ∈ L | (x, y, α) ∈ R}, y ∈ S 2 , defines a normalized L-fuzzy monotonic predicate m xR ∈ M [L] S 2 , and the correspondence R that takes each x to m xR is an isotone mapping S 1 → M [L] S 2 .On the other hand, similarly to the binary case, each isotone mapping R : S 1 → M [L] S 2 determines the L-ambiguous representation

For
an element d 0 ∈ D, we denote by η [L] D(d 0 ) the function D → L that sends each d ∈ D to 1 if d d 0 and to 0 otherwise.It is easy to see that η [L] D(d 0 ) ∈ M [L] D, and η [L] D(0) is a least element of M [L] D. It follows from [16, Lemma 1.1] that the mapping η [L] D : D → M [L] D is Scott continuous and lower continuous.Moreover, if D is a continuous semilattice with zero, then η [L] D is a zero-preserving semilattice morphism, hence we consider D as a subset of M [L] D. Proposition 13 ([14], 1.3)For each Scott continuous mapping ϕ : D → K from a domain with a bottom element to a continuous L-semimodule there is a unique Scott continuous affine extension Φ : M [L] D → K.It is linear if and only if ϕ preserves the bottom element.
For a given ϕ, we say that m : D → L is a precondition and m : D → L is a postcondition for each other w.r.t.ϕ, if, for all a ∈ D and b ∈ D , the "guaranteed" truth value m (b) is greater or equal to m(a) * ϕ(a)(b), i.e., to the result of modus ponens.Obviously, for an antitone function m : D → L, its strongest (least) postcondition Φ(m) in M [L] D is determined by the equality Φ(m)(b) = inf sup{m(a) * ϕ(a)(b ) | a ∈ D} | b ∈ D , b b , b ∈ D .
For monotonic predicates m : S → L, m : S → L let (m, m ) * P = m • m = sup{m(d) * P (d, d ) * m (d ) | d ∈ S, d ∈ S }.We use the second notation if P and * are easily guessed.It was proved that the introduced multiplication is infinitely distributive w.r.t.joins and uniform in the both arguments.Moreover, by Proposition 2.2 [14] M [L] S and M [L] S , together with the multiplication (−, −) * P : M [L] S × M [L] S → L, constitute a dual pair [6], i.e., the multiplication separates elements, both of M [L] S and of M [L] S , in the obvious sense.If to incompatible s ∈ S, s ∈ S monotonic predicates m and m assign truth values m(s) and m (s ) with non-zero conjunction m(s) * m (s ), then the predicates are also incompatible to this extent.The product (m, m ) *
S 1 the set xR = {y ∈ S 2 | (x, y) ∈ R} is non-empty and closed under directed sups in S 2 .Observe that (a) implies (x, 0 2 ) ∈ R for all x ∈ S. It also implies that xR in (b) is a lower set, hence due to (b) is Scott closed.Thus we can rearrange the requirements as follows, obtaining an equivalent definition: For continuous semilattices S 1 , S 2 with zeros, a binary relation R ⊂ S 1 × S 2 is an ambiguous representation of S 1 in S 2 if (a ) for all x ∈ S 1 the set xR = {y ∈ S 2 | (x, y) ∈ R} is non-empty and Scott closed (i.e., a directed complete lower set).(b ) for all y ∈ S 2 the set Ry Definition 2 Let S 1 , S 2 be continuous semilattices with zeros.An ambiguous representation of S 1 in S 2 is a binary relation R ⊂ S 1 × S 2 such that (a) if (x, y) ∈ R, x x in S 1 , and y y in S 2 , then (x , y ) ∈ R as well;(b) for all x ∈ 1 ⇒ * S 2 in SemPR * L and with R : S 1 ⇒ * S 2 in SemPR * L .The composition of R and Q is denoted by R * SemPR * L .Both pairwise compositions of these functors are isomorphic to the identity functors.
Proposition 12 The self-duality (−) : SemPR → SemPR extends to contravariant functors (−) : SemPR * L → SemPR * L and (−) : SemPR * L → and only if for all z z and α α we have α sup{ϕ(x)(y) * ψ(y)(z ) | y ∈ S 2 }.As the least upper bound of a set A in L is the limit of the directed net of the least upper bounds of the finite subsets of A, the latter condition is equivalent to existence, for all α α and z z, of y 1 , y 2 , . . ., y n ∈ S 2 such that ϕ