* algebra/catdef.spad.pamphlet (BinaryOperatorCategory): New.

	(BinaryOperation): Likewise.
	(IdempotentOperatorCategory): Likewise.
	(SemiGroupOperatorCategory): Likewise.
	(SemiGroupOperation): Likewise.
	(MonoidOperatorCategory): Likewise.
	(MonoidOperation): Likewise.
	* algebra/exposed.lsp.pamphlet: Expose them.
	* algebra/Makefile.in (oa_algebra_layer_0): Include IDEMOPC,
	SGPOPC, SPGOP, MONOPC, BINOPC.
	(axiom_algebra_layer_1): Include MONOP.

3 files changed, 153 insertions, 7 deletions

diff --git a/src/algebra/Makefile.in b/src/algebra/Makefile.in
index 7f30bb8e..0077bbde 100644
--- a/src/algebra/Makefile.in
+++ b/src/algebra/Makefile.in
@@ -1428,17 +1428,25 @@ $(OUT)/DSEXT.$(FASLEXT): $(OUT)/DIFFSPC.$(FASLEXT) $(OUT)/PDSPC.$(FASLEXT)
 $(OUT)/ORDTYPE.$(FASLEXT): $(OUT)/BASTYPE.$(FASLEXT)
 $(OUT)/ORDSTRCT.$(FASLEXT): $(OUT)/ORDTYPE.$(FASLEXT) $(OUT)/HOMOTOP.$(FASLEXT)
 
+$(OUT)/BINOPC.$(FASLEXT): $(OUT)/TYPE.$(FASLEXT)
+$(OUT)/BINOP.$(FASLEXT): $(OUT)/BINOPC.$(FASLEXT) $(OUT)/SETCAT.$(FASLEXT)
+$(OUT)/IDEMOPC.$(FASLEXT): $(OUT)/BINOPC.$(FASLEXT) $(OUT)/BASTYPE.$(FASLEXT)
+$(OUT)/SGPOPC.$(FASLEXT): $(OUT)/BINOPC.$(FASLEXT) $(OUT)/BASTYPE.$(FASLEXT)
+$(OUT)/SGPOP.$(FASLEXT): $(OUT)/SGPOPC.$(FASLEXT) $(OUT)/SETCAT.$(FASLEXT) \
+	$(OUT)/BINOP.$(FASLEXT)
+$(OUT)/MONOPC.$(FASLEXT): $(OUT)/SGPOPC.$(FASLEXT)
+
 oa_algebra_layer_0 = \
   AHYP     ATTREG   CFCAT    ELTAB    KOERCE   KONVERT  \
   KRCFROM  KVTFROM  IEVALAB  IEVALAB- EVALAB   EVALAB-  \
   RETRACT  RETRACT- SETCAT   SETCAT-  VOID     SEGCAT   \
-  MSYSCMD  FINITE   FINITE-                    OUT      \
+  MSYSCMD  FINITE   FINITE-  IDEMOPC           OUT      \
   PRIMCAT  PRINT    PTRANFN  SPFCAT   TYPE     UTYPE    \
-  BMODULE           BASTYPE  BASTYPE- STEP     LMODULE  \
+  BMODULE  SGPOPC   BASTYPE  BASTYPE- STEP     LMODULE  \
   RMODULE  ALGEBRA  ALGEBRA- SGROUP   SGROUP-  ABELSG   \
-  ABELSG-  ORDSET            OASGP    FILECAT  SEXCAT   \
+  ABELSG-  ORDSET   SGPOP    OASGP    FILECAT  SEXCAT   \
   MODULE   MODULE-  PID      OAGROUP  OAGROUP- OCAMON   \
-  OAMON    OAMON-   DIOID                               \
+  OAMON    OAMON-   DIOID    MONOPC                     \
   RNG      ORDFIN   LLINSET  RLINSET  LINSET   OAMONS   \
   MKBCFUNC MKRECORD MKUCFUNC DROPT1   PLOT1    ITFUN2   \
   ITFUN3   STREAM1  STREAM2  STREAM3  ANY1     SEGBIND2 \
@@ -1458,7 +1466,7 @@ oa_algebra_layer_0 = \
   DSEXT    DSEXT-   ORDTYPE  ORDTYPE- ORDSTRCT          \
   BOOLE    BOOLE-   REF      ALIST    PRIMARR  SRING    \
   INS      INS-     DIVRING  DIVRING- EUCDOM   EUCDOM-  \
-                    FPS      FPS-     RNS      RNS-     \
+  BINOPC   BINOP    FPS      FPS-     RNS      RNS-     \
   POLYCAT  POLYCAT- QFCAT    QFCAT-                     \
   INT      NNI      PI       SINT     SYMBOL   DFLOAT
 
@@ -1523,6 +1531,9 @@ $(OUT)/DMEXT.$(FASLEXT): $(OUT)/DSEXT.$(FASLEXT) $(OUT)/DIFFMOD.$(FASLEXT) \
 	$(OUT)/PDMOD.$(FASLEXT)
 $(OUT)/STREAM.$(FASLEXT): $(OUT)/LZSTAGG.$(FASLEXT)
 
+$(OUT)/MONOP.$(FASLEXT): $(OUT)/MONOPC.$(FASLEXT) $(OUT)/PAIR.$(FASLEXT) \
+	$(OUT)/SETCAT.$(FASLEXT)
+
 axiom_algebra_layer_1 = \
   ABELGRP  ABELGRP- ABELMON  ABELMON- ITUPLE   \
   CABMON   MONOID   MONOID-  RING     RING-     COMRING  \
@@ -1532,7 +1543,7 @@ axiom_algebra_layer_1 = \
   IDENT    OUTFORM  BINDING           BOOLEAN            \
   ORDRING                    FEVALAB  FEVALAB-  IARRAY1  \
   OSGROUP  MAYBE    DATAARY  PROPLOG  HOMOTOP   BYTEORD  \
-  FIELD    FIELD-   VECTCAT  VECTCAT- \
+  FIELD    FIELD-   VECTCAT  VECTCAT- MONOP              \
   PROPERTY ARITY    OPERCAT  OPERCAT- PAIR      STREAM   \
   RADCAT   RADCAT-  PDMOD    DMEXT    LZSTAGG   LZSTAGG-
 
diff --git a/src/algebra/catdef.spad.pamphlet b/src/algebra/catdef.spad.pamphlet
index 00930fb2..997b4bba 100644
--- a/src/algebra/catdef.spad.pamphlet
+++ b/src/algebra/catdef.spad.pamphlet
@@ -12,6 +12,126 @@
 \tableofcontents
 \eject
 
+\section{Binary operations}
+
+<<category BINOPC BinaryOperatorCategory>>=
+)abbrev category BINOPC BinaryOperatorCategory
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This is the category of all domains that implement binary operations.
+BinaryOperatorCategory(T: Type): Category == MappingCategory(T,T,T)
+
+@
+
+<<domain BINOP BinaryOperation>>=
+)abbrev domain BINOP BinaryOperation
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This domain implements binary operations.
+BinaryOperation(T: Type): Join(BinaryOperatorCategory T, SetCategory) with
+    binaryOperation: ((T,T) -> T) -> %
+      ++ \spad{binaryOperation f} constructs a binary operation value
+      ++ out of any homogeneous mapping of arity 2.
+  == (T,T) -> T add
+    binaryOperation f == per f
+    elt(f,x,y) == rep(f)(x,y)
+
+@
+
+
+\section{Idempotent operations}
+
+<<category IDEMOPC IdempotentOperatorCategory>>=
+)abbrev category IDEMOPC IdempotentOperatorCategory
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This is the category of all domains that implement idempotent operations.
+IdempotentOperatorCategory(T: BasicType): Category ==
+  BinaryOperatorCategory T with
+    assume idempotence ==
+      forall(f: %, x: T) . f(x,x) = x
+@
+
+
+\section{Semigroup operations}
+
+<<category SGPOPC SemiGroupOperatorCategory>>=
+)abbrev category SGPOPC SemiGroupOperatorCategory
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This is the category of all domains that implement semigroup operations
+SemiGroupOperatorCategory(T: BasicType): Category ==
+  BinaryOperatorCategory T with
+    assume associativity ==
+      forall(f: %, x: T, y:T, z: T) . f(f(x,y),z) = f(x,f(y,z))
+
+@
+
+<<domain SGPOP SemiGroupOperation>>=
+)abbrev domain SGPOP SemiGroupOperation
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This domain implements semigroup operations.
+SemiGroupOperation(T: BasicType): Public == Private where
+  Public == Join(SemiGroupOperatorCategory T,SetCategory) with
+    semiGroupOperation: ((T,T) -> T) -> %
+      ++ \spad{semiGroupOperation f} constructs a semigroup operation
+      ++ out of a binary homogeneous mapping known to be associative.
+  Private == BinaryOperation T
+
+@
+
+
+\section{Monoid operations}
+
+<<category MONOPC MonoidOperatorCategory>>=
+)abbrev category MONOPC MonoidOperatorCategory
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This is the category of all domains that implement monoid operations
+MonoidOperatorCategory(T: BasicType): Category ==
+  SemiGroupOperatorCategory T with
+    neutralValue: % -> T
+      ++ \spad{neutralValue f} returns the neutral value of the
+      ++ monoid operation \spad{f}.
+    assume neutrality ==
+      forall(f: %, x: T) .
+         f(x, neutralValue f) = x
+         f(neutralValue f, x) = x
+
+@
+
+<<domain MONOP MonoidOperation>>=
+)abbrev domain MONOP MonoidOperation
+++ Author: Gabriel Dos Reis
+++ Date Created: February 24, 2012
+++ Date Last Modified: February 24, 2012
+++ Description:
+++   This domain implements monoid operations.
+MonoidOperation(T: BasicType): Public == Private where
+  Public == Join(MonoidOperatorCategory T,SetCategory) with
+    monoidOperation: ((T,T) -> T, T) -> %
+      ++ \spad{monoidOperation(f,e)} constructs a operation from
+      ++ the binary mapping \spad{f} with neutral value \spad{e}.
+  Private == Pair((T,T) -> T,T) add
+    monoidOperation(f,e) == per pair(f,e)
+    elt(op,x,y) == first(rep op)(x,y)
+    neutralValue op == second rep op
+@
+
+
 \section{Linear sets}
 
 <<category LLINSET LeftLinearSet>>=
@@ -2065,7 +2185,7 @@ VectorSpace(S:Field): Category ==  Module(S) with
 <<license>>=
 --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
 --All rights reserved.
---Copyright (C) 2007-2010, Gabriel Dos Reis.
+--Copyright (C) 2007-2012, Gabriel Dos Reis.
 --All rights reversed.
 --
 --Redistribution and use in source and binary forms, with or without
@@ -2099,6 +2219,14 @@ VectorSpace(S:Field): Category ==  Module(S) with
 <<*>>=
 <<license>>
 
+<<category BINOPC BinaryOperatorCategory>>
+<<domain BINOP BinaryOperation>>
+<<category IDEMOPC IdempotentOperatorCategory>>
+<<category SGPOPC SemiGroupOperatorCategory>>
+<<domain SGPOP SemiGroupOperation>>
+<<category MONOPC MonoidOperatorCategory>>
+<<domain MONOP MonoidOperation>>
+
 <<category BASTYPE BasicType>>
 <<category ORDTYPE OrderedType>>
 <<domain ORDSTRCT OrderedStructure>>
diff --git a/src/algebra/exposed.lsp.pamphlet b/src/algebra/exposed.lsp.pamphlet
index 0c8523cf..85f460f4 100644
--- a/src/algebra/exposed.lsp.pamphlet
+++ b/src/algebra/exposed.lsp.pamphlet
@@ -70,6 +70,7 @@
   (|BasicOperator| . BOP)
   (|BasicOperatorFunctions1| . BOP1)
   (|BinaryExpansion| . BINARY)
+  (|BinaryOperation| . BINOP)
   (|BinarySearchTree| . BSTREE)
   (|BinaryTournament| . BTOURN)
   (|BinaryTree| . BTREE)
@@ -260,6 +261,7 @@
   (|MergeThing| . MTHING)
   (|ModularDistinctDegreeFactorizer| . MDDFACT)
   (|ModuleOperator| . MODOP)
+  (|MonoidOperation| . MONOP)
   (|MonoidRingFunctions2| . MRF2)
   (|MoreSystemCommands| . MSYSCMD)
   (|MPolyCatFunctions2| . MPC2)
@@ -386,6 +388,7 @@
   (|SegmentBinding| . SEGBIND)
   (|SegmentBindingFunctions2| . SEGBIND2)
   (|SegmentFunctions2| . SEG2)
+  (|SemiGroupOperation| . SGPOP)
   (|SequenceAst| . SEQAST)
   (|Set| . SET)
   (|Signature| . SIG)
@@ -476,6 +479,7 @@
   (|BagAggregate| . BGAGG)
   (|BasicType| . BASTYPE)
   (|BiModule| . BMODULE)
+  (|BinaryOperatorCategory| . BINOPC)
   (|BinaryRecursiveAggregate| . BRAGG)
   (|BinaryTreeCategory| . BTCAT)
   (|BitAggregate| . BTAGG)
@@ -553,6 +557,7 @@
   (|HomogeneousAggregate| . HOAGG)
   (|HomotopicTo| . HOMOTOP)
   (|HyperbolicFunctionCategory| . HYPCAT)
+  (|IdempotentOperatorCategory| . IDEMOPC)
   (|IndexedAggregate| . IXAGG)
   (|IndexedDirectProductCategory| . IDPC)
   (|InnerEvalable| . IEVALAB)
@@ -579,6 +584,7 @@
   (|Monad| . MONAD)
   (|MonadWithUnit| . MONADWU)
   (|Monoid| . MONOID)
+  (|MonoidOperatorCategory| . MONOPC)
   (|MonogenicAlgebra| . MONOGEN)
   (|MonogenicLinearOperator| . MLO)
   (|MultiDictionary| . MDAGG)
@@ -649,6 +655,7 @@
   (|SegmentCategory| . SEGCAT)
   (|SegmentExpansionCategory| . SEGXCAT)
   (|SemiGroup| . SGROUP)
+  (|SemiGroupOperatorCategory| . SGPOPC)
   (|SemiRing| . SRING)
   (|SetAggregate| . SETAGG)
   (|SetCategory| . SETCAT)