Extremal Elements

Greatest (Max)

$\textbf{Greatest} \; A \; R \; Y \; x$

---

$\textbf{Preorder} \; A \; R$

$\textbf{Includes} \; A \; Y$

$x \in Y$

$\textbf{Above} \; A \; R \; A \; x$

---

Notation.

1. $\textbf{Greatest} \; A \; R \; Y \; x$ can be written $\textbf{Greatest} \; Y \; x$ when $A$ and $R$ are clear from the context.

2. $\textbf{Greatest} \; Y \; x$ can be written $x = \textbf{max} \; Y$.

pred Greatest(A: set univ, R: univ->univ, Y: set univ, x: univ) {
  Preorder[A,R]
  Includes[A,Y]
  x in Y
  Above[A,R,A,x]
}

$\textbf{Greatest} \; A \; R \; Y := \{\; x : \textbf{Greatest} \; A \; R \; Y \; x\;\}$

---

Notation.

1. $\textbf{Greatest} \; A \; R \; Y$ can be written $\textbf{Greatest} \; Y$ when $A$ and $R$ are clear from the context.

2. $\textbf{Greatest} \; Y$ can be written $\textbf{max} \; Y$.

---

fun Greatest(A: set univ, R: univ->univ, Y: set univ) : set univ {
  { x: Y | Greatest[A,R,Y,x] }
}

Join (Supremum)

$\textbf{Join} \; A \; R \; S \; x$

---

$\textbf{Above} \; A \; R \; S \; x$

$\forall(y : \textbf{Above} \; A \; R \; S \; y : R.x.y)$

---

pred Join(A: set univ, R: univ->univ, S: set univ, x: univ) {
  Above[A,R,S,x]
  all y: univ | Above[A,R,S,y] implies x->y in R
}

$\textbf{Join} \; A \; R \; S$

---

$\{ \; x : \textbf{Join} \; A \; R \; S \; x \; \}$

---

fun Join(A: set univ, R: univ->univ, S: set univ) : set univ {
  { x: univ | Join[A,R,S,x] }
}

Extended Join

$\textbf{JoinExt} \; A \; R \; Z \; Y \; x$

---

$\textbf{Includes} \; Z \; A$

$\textbf{Includes} \; Y \; Z$

$x \in Z$

$\forall(z \in Z :: \textbf{Below} \; A \; R \; Y \; z \equiv R.z.x)$

---

Notation.

1. $\textbf{JoinExt} \; Z \; Y \; x$ abbreviates $\textbf{JoinExt} \; A \; R \; Z \; Y \; x$ when $A$ and $R$ are clear from the context.
2. $\textbf{JoinExt} \; Z \; Y \; x$ is denoted by $x = \sqcup_Z Y$.

---

pred JoinExt(A: set univ, R: univ->univ, Z,Y: set univ, x: univ) {
  Includes[Z,A]
  Includes[Y,Z]
  x in Z
  all z: Z | Above[A,R,Y,z] iff x->z in R
}

$\textbf{JoinExt} \; A \; R \; Z \; Y := \{\; x : \textbf{JoinExt} \; A \; R \; Z \; Y \; x\;\}$

---

Notation.

1. $\textbf{JoinExt} \; Z \; Y$ abbreviates $\textbf{JoinExt} \; A \; R \; Z \; Y$ when $A$ and $R$ are clear from the context.
2. $\textbf{JoinExt} \; Z \; Y$is denoted by $\sqcup_Z Y$.

---

fun JoinExt(A: set univ, R: univ->univ, Z,Y: set univ) : set univ {
  { x: Z | JoinExt[A,R,Z,Y,x] }
}

Least (Min)

$\textbf{Least} \; A \; R \; Y \; x$

---

$\textbf{Preorder} \; A \; R$

$\textbf{Includes} \; Y \; A$

$x \in Y$

$\textbf{Below} \; A \; R \; A \; x$

---

Notation

1. $\textbf{Least} \; A \; R \; Y \; x$ can be abbreviated by $\textbf{Least} \; Y \; x$ when $A$ and $R$ are clear from the context.
2. $\textbf{Least} \; Y \; x$ can be written $x = \textbf{min} \; Y$.

---

pred Least(A: set univ, R: univ->univ, Y: set univ, x: univ) {
  Preorder[A,R]
  Includes[Y,A]
  x in Y
  Below[A,R,A,x]
}

$\textbf{Least} \; A \; R \; Y := \{ \; x : \textbf{Least} \; A \; R \; Y \; x \; \}$

---

Notation.

1. $\textbf{Least} \; A \; R \; Y$ can be written $\textbf{Least} \; Y$when $A$ and $R$ are clear from the context.
2. $\textbf{Least} \; Y$ can be written $\textbf{min} \; Y$.

---

fun Least(A: set univ, R: univ->univ, Y: set univ) : set univ {
  { x: Y | Least[A,R,Y,x] }
}

Meet (Infimum)

$\textbf{Meet} \; A \; R \; S \; x$

---

$\textbf{Below} \;A \; R \; S \; x$

$\forall(y : \textbf{Below} \;A \; R \; S \; y : R.y.x)$

---

Notation.

1. $\textbf{Meet} \; S \; x$ abbreviates $\textbf{Meet} \; A \; R \; S \; x$ when $A$ and $R$​ are clear from the context.
2. $\textbf{Meet} \; S \; x$ is abbreviated by $x = \sqcap \; S$

---

pred Meet(A: set univ, R: univ->univ, S: set univ, x: univ) {
  Below[A,R,S,x]
  all y: univ | Below[A,R,S,y] implies y->x in R
}

$\textbf{Meet} \; A \; R \; S := \{ \; x : \textbf{Meet} \; A \; R \; S \; x \; \}$

---

Notation.

1. $\textbf{Meet} \; S$ abbreviates $\textbf{Meet} \; A \; R \; S$ when $A$ and $R$​ are clear from the context.
2. $\textbf{Meet} \; S$ is abbreviated by $\sqcap \; S$

---

fun Meet(A: set univ, R: univ->univ, S: set univ) : set univ {
  { x : univ | Meet[A,R,S,x] }
}

Extended Meet

$\textbf{MeetExt} \; A \; R \; Z \; Y \; x$

---

$\textbf{Includes} \; Z \; A$

$\textbf{Includes} \; Y \; Z$

$x \in Z$

$\forall(z \in Z :: \textbf{Below} \; A \; R \; Y \; z \equiv R.z.x)$

---

Notation.

1. $\textbf{MeetExt} \; Z \; Y \; x$ abbreviates $\textbf{MeetExt} \; A \; R \; Z \; Y \; x$ when $A$ and $R$ are clear from the context.
2. $\textbf{MeetExt} \; Z \; Y \; x$ is denoted by $x = \sqcap_Z Y$.

---

pred MeetExt(A: set univ, R: univ->univ, Z,Y: set univ, x: univ) {
  Includes[Z,A]
  Includes[Y,Z]
  x in Z
  all z: Z | Below[A,R,Y,z] iff z->x in R
}

$\textbf{MeetExt} \; A \; R \; Z \; Y := \{\; x : \textbf{MeetExt} \; A \; R \; Z \; Y \; x\;\}$

---

Notation.

1. $\textbf{MeetExt} \; Z \; Y$ abbreviates $\textbf{MeetExt} \; A \; R \; Z \; Y$ when $A$ and $R$ are clear from the context.
2. $\textbf{MeetExt} \; Z \; Y$is denoted by $\sqcap_Z Y$.

---

fun MeetExt(A: set univ, R: univ->univ, Z,Y: set univ) : set univ {
  { x: univ | MeetExt[A,R,Z,Y,x] }
}