Corrections to June/1992 Version of TEX-ed “Domains”

Notation. In the following list, figures/diagrams/tables (and their captions) are not included in the counting of the number of lines otherwise stated, while display-style mathematical formulae and title-lines of chapters/sections are included in the counting. A negative line-number means the number of lines counted from the end of each page. Each misprinted part and the corresponding correction are enclosed with a pair of the text-typewriter-style double quotation symbols (").

Acknowledgements. The editors of the TEX edition wish to deepest thanks to Professor John C. Mitchell, who informed them that there remained many misspellings in the first version and strongly advised them to use some spell checker, and also have kindly spent his important time to pass the TeX source files to spell-checkers and told them the results with the original author’s comments and directions concerning misspellings and the original (British) spellings in the notes. They also thank Marcelo Fiore at Univ. of Edinburgh and Daisuke Suzuki at Univ. of Tokyo for notifying them of many misprints.

November, 1992 the editors^{†}

  † Hidetaka Kondoh (kondoh@harl.hitachi.co.jp),
    Yugo Kashiwagi (kasiwagi@crl.hitachi.co.jp).

[p. i, l. 23] Replace "Funstion Spaces" by "Function Spaces".

Notes to the TEX Edition and Acknowledgements

[p. iii, l. 7] Replace "edtion" by "edition".

Chapter 1

[p. 1, l. 1] Replace "Continusous" by "Continuous". [p. 1, l. −5] Replace "(let f(x) be e in e)" by "(let f_{1}(x) be e in e)". [p. 3, l. 10] Replace "w ≠ ⊥" by "w(⊥) ≠ ⊥". [p. 4, l. 15] Replace "Let we" by "Let us". [p. 5, l. 7] Replace "includsion" by "inclusion". [p. 5, l. −20 (Definition 1)] Replace "contiuous" by "continuous". [p. 5, l. −18] Replace "id_{D}" by "id_{D}". [p. 6, l. −22] Replace "holds an" by "holds and". [p. 8, l. −16] Replace "if b_{1} then" by "if b then". [p. 9, l. 2] Replace "thier" by "their". [p. 9, l. 19] Replace "Given" by "Give". [p. 9, l. 27] Replace "⟨x, y⟩ ⊆ ⟨u, v⟩" by "⟨x, y⟩ ⊑ ⟨u, v⟩". [p. 9, l. −10] Replace "Tapes:" by "Tapes;". [p. 9, l. −7] Replace "_{⊓_{D}} f(x_{n})" by "_{⊓_{E}} f(x_{n})". [p. 9, l. −3] Replace "Fun" by "Pred". [p. 9, l. −3] Replace "minimilisation" by "minimalisation". [p. 9, l. −2] Replace "minimilisation" by "minimalisation". [p. 10, l. 9] Replace "{0}^{ω}" by "{0}^{∞}". [p. 10, l. −14] Replace "_{⨆m}(_{⨆n} a_{m,n}) =" by "_{⨆m}(_{⨆n} d_{m,n}) =". [p. 10, l. −13] Replace "_{⨆m,n} a_{m,n} = _{⨆n} a_{n,n}" by "_{⨆m,n} d_{m,n} = _{⨆n} d_{n,n}". [p. 10, l. −8] Replace "reserved" by "recovered". [p. 10, l. −7] Replace "that D, E are" by "that if D, E are".

Chapter 2

[p. 11, l. 25] Replace "ans so on" by "and so on". [p. 11, l. −9∼8] Replace "determinded" by "determined". [p. 11, l. −3] Replace "f_{i}: F → D" by "f_{i}: F → D_{i}". [p. 12, l. 4] Replace "⟨f_{0}(x), …, f_{n}(x)⟩" by "⟨f_{0}(x), …, f_{n−1}(x)⟩". [p. 12, l. 10] Replace "⟨f_{0}, …, f_{n}⟩" by "⟨f_{0}, …, f_{n−1}⟩". [p. 12, l. 11] Replace "a unique morphisms" by "a unique morphism". [p. 12, l. 17] Replace "instead of more" by "instead of the more".

[p. 12, l. 23] Replace "We call for" by "We call the". [p. 12, l. −9] Replace "Now let h be" by "Now let h′ be". [p. 13, l. 10] Replace "α[e_{n−1}; x_{0}, …, x_{n−1}]" by "α[e_{m−1}; x_{0}, …, x_{n−1}]". [p. 13, l. 13] Replace "f(x_{0}, …, x_{i−1}, a, x_{i+1}, …, x_{n−1})" by "f(x_{0}, …, x_{i−1}, d_{i}, x_{i+1}, …, x_{n−1})". [p. 14, l. 14] Replace "flat domain" by "flat domains". [p. 15, l. 2] Replace "Btree" by "Btrees". [p. 15, l. 4] Replace "Btree" by "Btree". [p. 15, l. 5] Replace "t ⊑ t′_{1}" by "t_{1} ⊑ t′_{1}". [p. 15, l. 6] Replace "⨆_{n} T_{n} ⊑ ⨆_{n} t′_{n}" by "⨆_{n} t_{n} ⊑ ⨆_{n} t′_{n}". [p. 15, l. 8] Replace "mktree: Btree × Btree → Btree" by "mktree: Btrees × Btrees → Btrees". [p. 15, l. 8] Replace "val: Btree → N" by "val: Btrees → N". [p. 15, l. 9] Replace "ltree, rtree: Btree → Btree" by "ltree, rtree: Btrees → Btrees". [p. 15, l. 9] Replace "istip: Btree → T" by "istip: Btrees → T". [p. 15, l. −12] Replace "∏: CPO → CPO" by "∏: CPO^{n} → CPO". [p. 15, l. −3] Replace "f(x) ⊑_{E} f(y)" by "f(x) ⊑_{E} g(x)". [p. 16, l. −13] Replace "socontinuous" by "so continuous". [p. 17, l. 10] Replace "expressoin" by "expression". [p. 17, l. 12] Replace "contiuous" by "continuous". [p. 17, l. 16] Replace "π_{n−1} ∘ π_{0}" by "π_{m−1} ∘ π_{0}". [p. 17, l. 16] Replace "⟨π_{0}, …, π_{n−1}⟩" by "⟨π_{0}, …, π_{m−1}⟩". [p. 17, l. 20] Replace "expression" by "expressions". [p. 17, l. 31] Replace "x1: D_{1}" by "x_{1}: D_{1}". [p. 18, l. 3] Replace "⟨v, d⟩)(v)" by "⟨v, d⟩))(v)". [p. 18, l. 8] Replace "Currying (Curry)" by "Currying (Curry)". [p. 18, l. 13] Replace "f: A′ → A" by "f: B → A". [p. 18, l. 13] Replace "f: A → A′" by "f: A → B". [p. 18, l. 20] Replace "where extended" by "when extended". [p. 18, l. 22] Replace "fix" by "fix". [p. 18, l. −9] Replace "minimilisation" by "minimalisation". [p. 18, l. −9] Replace "p_{⊥}: N_{⊥}^{2} → T" by "p_{⊥}: N_{⊥}^{2} → T_{⊥}". [p. 18, l. −2] Replace "sum(f, n − 1)))" by "sum(f, n − 1))". [p. 20, l. 4] Replace "∏_{i} P_{i<n}" by "∏_{i<n} P_{i}". [p. 20, l. 12] Replace "which is defied" by "which is defined". [p. 20, l. −21] Replace "comands" by "commands". [p. 20, l. −15] Replace "Fenv" by "FEnv". [p. 20, l. −15] Replace "(V^{n} → V)" by "(V^{j} → V)".

[p. 20, l. −8] Replace "E[[e_{0}]](ρ[E[[e_{1}]]" by "E[[e_{1}]](ρ[E[[e_{0}]]". [p. 21, l. 12] Replace "isomorphism" by "isomorphisms". [p. 21, l. 15] Replace "E_{n−1}" by "E_{m−1}". [p. 21, l. 19] Replace "In it" by "Is it". [p. 21, l. 24] Replace "f_{i}: D → E′_{i}" by "f_{i}: D → E_{i}". [p. 22, l. 21] Replace "minimilisation" by "minimalisation". [p. 22, l. −16] Replace "Bekic" by "Bekic̆". [p. 23, l. −3] Replace "D →^{θ}_{⊥} P" by "C →^{θ}_{⊥} P". [p. 24, l. −12] Replace "E[[[letrec f_{n}(x_{0}, …, x_{n−1})" by "E[[letrec f_{n}(x_{0}, …, x_{n−1})".

Chapter 3

[p. 25, l. 8] Replace "we wil" by "we will". [p. 25, l. 11] Replace "teminate" by "terminate". [p. 25, l. 18] Replace "(x)_{j}" by "(x)_{i}". [p. 25, l. −18] Replace "function spaces" by "function space". [p. 26, l. 8] Replace "g: ⨂_{i<n} →_{⊥}E" by "g: ⨂_{i<n} D_{i} →_{⊥} E". [p. 26, l. 9] Replace "htat" by "that". [p. 26, l. 11] Replace "f: ∏_{i<n} D_{i} →_{⊥} E." by "∀f: ∏_{i<n} D_{i} →_{⊥} E.". [p. 26, l. 12] Replace "g: ⨂_{i<n} D_{i} →_{⊥} E." by "∀g: ⨂_{i<n} D_{i} →_{⊥} E.". [p. 26, l. 17] Replace "x_{0}: D_{0}, …, D_{n−1}" by "x_{0}: D_{0}, …, x_{n−1}: D_{n−1}". [p. 26, l. 18] Replace "x_{0} ⊗ ⋯ x_{n−1}" by "x_{0} ⊗ ⋯ ⊗ x_{n−1}". [p. 26, l. 21] Replace "π_{n−1}(x_{0}⊗, …, x_{n−1})" by "π_{n−1}(x_{0} ⊗ ⋯ ⊗ x_{n−1})". [p. 26, l. −6] Replace "Chapter 2 in" by "Chapter 2 is". [p. 27, l. 5 (the math. formula No. (i))] Replace "id_{⊗D_{i}}" by "id_{⨂ D_{i}}". [p. 27, l. −10] Replace "repharased" by "rephrased". [p. 28, l. 4] Replace "take take" by "take". [p. 28, l. 4] Replace "I_{i}: Op_{i} → (V\underline{j} →_{⊥} V)" by "I_{i}: Op_{i} → (V\underline{i} →_{⊥} V)". [p. 28, l. 14 (the semantic equation (vi)_{j})] Replace "[(w)_{j−1}/x_{j−1}];" by "[(w)_{j−1}/x_{j−1}],". [p. 28, l. 16] Replace "FEnv" by "IEnv". [p. 29, l. −7] Replace "in_{i}: ∑_{i<n} D_{i} →_{⊥} T" by "is_{i}: ∑_{i<n} D_{i} →_{⊥} T". [p. 30, l. 5] Replace "∑: CPO_{⊥}^{n} → CPO_{⊥}^{n}" by "∑: CPO_{⊥}^{n} → CPO_{⊥}". [p. 30, l. −7] Replace "{⟨o, d⟩ | d ∈ D}" by "{⟨o, d⟩ | d ∈ D} ∪ {⊥}". [p. 32, l. 17] Replace "∏_{i<n}(D →_{⊥} E)" by "∏_{i<n}(D →_{⊥} E_{i})". [p. 33, l. 3] Replace "cons: D^{∗} ⊗ D^{∗} → D^{∗}" by "cons: D ⊗ D^{∗} → D^{∗}". [p. 33, l. −5] Replace "(D × E)_{⊥};" by "(D × E)_{⊥},". [p. 34, the diagram] Replace "∑ (D^{∗})_{⊥}" by "∑ (D_{i})_{⊥}". [p. 34, l. 6] Replace "where the element" by "where the elements".

Chapter 4

[p. 35, l. 15] Replace "procedures" by "procedure". [p. 35, l. 18] Replace "assigning the" by "assigning to". [p. 35, l. 22] Replace "equatiies" by "equalities". [p. 35, l. 23] Replace "simulatneous" by "simultaneous". [p. 35, l. 25 (the math. formula No. (2))] Replace "Proc_{⊥} → C" by "Proc_{⊥} →_{⊥} C". [p. 35, l. −5] Replace "in analogous" by "is analogous". [p. 36, l. 1] Replace "n monotonic function" by "a monotonic function". [p. 36, l. 2] Replace "F f: fd → F e" by "F f: F d → F e". [p. 36, l. 34] Replace "save" by "says". [p. 37, l. 6] Replace "in another" by "is another". [p. 37, l.14] Replace "i^{L}" by "i^{R}". [p. 37, l.14] Replace "j^{R}" by "j^{L}". [p. 38, l. 19] Replace "(D →^{E} E) → (D → E)" by "(D →^{E} E) = (D → E)". [p. 39, l. 15] Replace "(i < n)" by "(i < m)". [p. 39, l. 15] Replace "(i < m)" by "(i < n)". [p. 40, l. l. −6] Replace "satsfy" by "satisfy". [p. 40, the lower diagram] The direction of the arrow with the label ρ_{n} is not downward-leftward but upward-rightward. [p. 40, the diagram] Replace "D" as the name of the lowest object by "D′". [p. 41, l. 11] Replace "render" by "reader". [p. 41, l. −11] Replace "ρ′_{m+1} ∘ f_{m} ∘ f_{m}^{R} ∘ ρ_{m+1}" by "ρ′_{m+1} ∘ f_{m} ∘ f_{m}^{R} ∘ ρ_{m+1}^{R}". [p. 41, l. −9] Replace "in" by "is". [p. 42, l. 16] Replace "embeding" by "embedding". [p. 42, l. 21] Replace "ω-chein" by "ω-chain". [p. 42, l. 21] Replace "unversal" by "universal". [p. 44, l. −6] Replace "ix ⊒ y" by "ix ⊑ y". [p. 45, l. −2] Replace "diagrams" by "diagram". [p. 46, l. 6] Replace "CPO^{E} too" by "CPO^{E} then it is universal in CPO_{⊥} too". [p. 46, l. 10] Replace "ω-chain" by "ω-chains". [p. 46, l. 11] Replace "in" by "is".

Chapter 5

[p. 49, l. 9] Replace "as before and" by "as before, and". [p. 50, l. 8] Replace "domain fo" by "domain of". [p. 50, l. −16] Replace "clause" by "class". [p. 51, l. 6] Replace "accomodate" by "accommodate".

[p. 51, l. 17] Replace "particular tedious" by "particularly tedious". [p. 52, l. 19] Replace "⨆_{m} F (ρ_{m}) ∘ ρ_{m}^{R}" by "⨆_{m} F (ρ_{m}) ∘ ρ_{m+1}^{R}". [p. 53, l. 4] Replace "existance" by "existence". [p. 53, the third (triangular) diagram] Replace "ρ′_{m+1}" as the label of the downward arrow by "h". [p. 54, l. −21] Replace "h(n + 1) = α′(h(n))" by "h(n + 1) = α′(h(n)))". [p. 56, l. −4∼3] Replace "and so g ∘ α = β ∘ T g and so g ∘ α = β ∘ T g" by "and so g ∘ α = β ∘ T g". [p. 57, l. 10] Replace "axiomtisations" by "axiomatisations". [p. 57, l. 16] Replace "iszerro(n)" by "iszero(n)". [p. 58, l. −13] Replace "in_{0}(x)" by "in_{0}(σ)". [p. 58, l. −5] Replace "∀o: O∀r: R." by "∀o: O.∀r: R.".

Chapter 6

[p. 59, l. 9] Replace "d ⊑ x_{n}" by "d ⊑ _{⨆n} x_{n}". [p. 59, l. 9] Replace "imples" by "implies". [p. 59, l. −2] Replace "pints" by "points". [p. 60, l. 12] Replace "ω-finte" by "ω-finite". [p. 60, l. 15] Replace "st:" by "s.t.:". [p. 63, l. −5] Replace "at" by "s.t.". [p. 64, l. 2] Replace "B ⊆ B_{D}" by "B ⊆ B_{D}". [p. 65, l. −9] Replace "catefory" by "category". [p. 66, l. 8] Replace "(D_{⊥}" by "D_{⊥}". [p. 66, l. 16] Replace "Lemma." by "Lemma 2.". [p. 66, l. −11] Replace "defition" by "definition". [p. 66, l. −3] Replace "B_{Fix_{F}} " by "B_{Fix_{f}} ". [p. 67, l. −4] The sentence "There is also an idea of a subbasis." is not a part of Exercise 3. [p. 67, l. −2] Replace "Lemma 2" by "Lemma 3". [p. 68, l. 3] Replace "a lub, g" by "a lub, g". [p. 68, l. 8] Replace "ω-algbraic" by "ω-algebraic". [p. 68, l. 10] Replace "Lemma 2" by "Lemma 3". [p. 68, l. 14] Replace "{d_{i} ⇒ e_{i} | 1 ≤ i ≤ n}" by "↑{d_{i} ⇒ e_{i} | 1 ≤ i ≤ n}". [p. 68, l. 17] Replace "x ⊆ D × E" by "X ⊆ D × E". [p. 68, l. 24] Replace "X ⊆ D ⊗ E" by "X ⊆ D ⊗ E". [p. 68, l. 25] Replace "(⨆_{D} π_{1}(X)))" by "(⨆_{E} π_{1}(X))". [p. 68, l. 28] Replace "consistentcompleteness" by "consistent completeness".

Chapter 7

[p. 70, l. 5] Replace "funciton" by "function". [p. 70, l. 17] Replace "ϵ_{m} ↑ ϵ_{m′})" by "ϵ_{m} ↑ ϵ_{m′}". [p. 70, l. 37] Replace "β = c" by "β = ϵ". [p. 71, l. 11] Replace "⟨a, b⟩" by "⟨a, b⟩:". [p. 72, l. 1] Replace "idex" by "index". [p. 72, l. 7] Replace "V_{ω}, T^{ω}" by "V_{ω}, Trees". [p. 72, 22] Replace both two occurrences of "ϵ_{f(m)}" by "ϵ_{f(n)}". [p. 73, l. −14] Replace "discussionof" by "discussion of". [p. 75, l. 2] Replace "Left" by "Let". [p. 77, l. 23] Replace "ϵ × ϵ′" by "ϵ ⊗ ϵ′". [p. 77, l. 23] Replace "begin" by "being". [p. 78, l. 10] Replace "defineable" by "definable". [p. 79, l. −16] Replace "… ϵ_{k}, ϵ′_{k} …" by "… ϵ_{k} …, … ϵ′_{k} …".

Chapter 8

[p. 84, l. 1] Replace "→^{x}" by "→^{×}". [p. 84, l. 4] Replace "f^{†} ∘ x" by "f^{†} ∘ ×". [p. 84, l. 4] Replace "x = {·}^{†_{2}}" by "× = {·}^{†_{2}}". [p. 84, l. 12] Replace "recular" by "regular". [p. 84, l. 17] Replace "C → (S → S_{⊥})" by "C: Com → (S → S_{⊥})". [p. 84, l. 18] Replace "A by" by "R via". [p. 85, the transition diagram] Replace the right occurrence of "⟨2, “off”⟩" by "⟨3, “off”⟩". [p. 85, the transition diagram] Replace the right occurrence of "⟨2, “on”⟩" by "⟨3, “on”⟩". [p. 87, l. −12] Replace "ω(m)" by "Ω(m)". [p. 87, l. −7] Replace "mulitplicative" by "multiplicative". [p. 90, l. 3] Replace "B_{X}" by "B_{x}". [p. 91, l. 17] Replace "conludes" by "concludes". [p. 91, l. −1] Replace "d ∈ D" by "d ∈ D". [p. 92, l. 2] Replace "multilicative" by "multiplicative". [p. 92, l. 5] Replace "realtional" by "relational". [p. 94, l. −5] Replace "S_{⊥}" by "N_{⊥}". [p. 95, l. 3] Replace "attatched" by "attached". [p. 95, l. −6] Replace "comunication" by "communication". [p. 96, l. 6] Replace "f(n)_{2} ⊗ ∥r_{2}" by "f(n)_{2}∥r_{2}". [p. 97, l. 15] Replace "R[[α!e]]" by "R[[α!e]](σ)".

[p. 97, l. 23] Replace "denotatnional" by "denotational". [p. 97, l. −10] Replace "meaing" by "meaning". [p. 98, l. −19] Replace "powerdomain" by "powerdomains". [p. 100, l. 13] Replace "un" by "in". [p. 100, l. 18] Replace "diaram" by "diagram". [p. 101, l. 10] Replace "This is" by "This". [p. 104, l. −20] Replace "contable" by "countable". [p. 104, l. −17] Replace "a increasing" by "an increasing". [p. 105, l. −13] Replace "line" by "like". [p. 107, l. 17] Replace "lim_{m} x_{n} = x" by "lim_{n} x_{n} = x". [p. 107, l. 23] Replace "∀_{∞}m.b ⋢ y_{m})" by "∀_{∞}m.b ⋢ y_{m}". [p. 107, l. 26] Replace "ininite" by "infinite". [p. 107, l. 31] Replace "J (D)" by "F(D)". [p. 108, l. 21] Replace "resticted" by "restricted". [p. 109, l. 17] Replace "⋃_{P} χ" by "⋃_{P}χ".

[p. 109, l. −8] Replace "x ⊑_{S} Y " by "X ⊑_{S} Y ". [p. 110, l. −15] Replace "coplete" by "complete". [p. 110, l. −11] Replace "desctiption" by "description". [p. 111, l. −18] Replace "⨆_{S} X" by "⨆_{S}X".

[p. 111, l. −17] Replace "exest" by "exist". [p. 112, l. −3] Replace "Exerise 89" by "Exercise 89". [p. 113, l. 12] Replace "funcions" by "functions". [p. 113, l. −7] Replace "mutifunctions" by "multifunctions". [p. 115, l. 10] Remove two occurrences of "(missing in the original text)". These seem to be intentionally left blank by Professor Plotkin. [p. 115, l. 13] Replace "exteision" by "extension".